Compiled and solved problems in geometry and trigonometry,F.Smaradanhe

Florentin Smarandache
Compiled and Solved Problems
in Geometry and Trigonometry

255 Compiled and Solved Problems in Geometry and Trigonometry
1
FLORENTIN SMARANDACHE
255 Compiled and Solved Problems
in Geometry and Trigonometry
(from Romanian Textbooks)
Educational Publisher
2015

2
Peer reviewers:
Prof. Rajesh Singh, School of Statistics, DAVV, Indore (M.P.), India.
Dr. Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences,
Beijing 100190, P. R. China.
Mumtaz Ali, Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000,
Pakistan
Prof. Stefan Vladutescu, University of Craiova, Romania.
Said Broumi, University of Hassan II Mohammedia, Hay El Baraka Ben M'sik,
Casablanca B. P. 7951, Morocco.
E-publishing, Translation & Editing:
Dana Petras, Nikos Vasiliou
AdSumus Scientific and Cultural Society, Cantemir 13, Oradea, Romania
Copyright:
Florentin Smarandache 1998-2015
Educational Publisher, Chicago, USA
ISBN: 978-1-59973-299-2

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Table of Content
Explanatory Note..................................................................................................................................................... 4
Problems in Geometry (9th
grade)................................................................................................................... 5
Solutions ...............................................................................................................................................................11
Problems in Geometry and Trigonometry .................................................................................................38
Solutions ...............................................................................................................................................................42
Other Problems in Geometry and Trigonometry (10th
grade)..........................................................60
Solutions ...............................................................................................................................................................67
Various Problems...................................................................................................................................................96
Solutions ...............................................................................................................................................................99
Problems in Spatial Geometry...................................................................................................................... 108
Solutions ............................................................................................................................................................ 114
Lines and Planes ................................................................................................................................................. 140
Solutions ............................................................................................................................................................ 143
Projections............................................................................................................................................................. 155
Solutions ............................................................................................................................................................ 159
Review Problems................................................................................................................................................. 174
Solutions ............................................................................................................................................................ 182

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Explanatory Note
This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de
Geometrie şi Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and
includes problems of 2D and 3D Euclidean geometry plus trigonometry, compiled and
solved from the Romanian Textbooks for 9th and 10th grade students, in the period
1981-1988, when I was a professor of mathematics at the "Petrache Poenaru" National
College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan Lyoussi in Sefrou (Morocco),
then at the "Nicolae Balcescu" National College in Craiova and Dragotesti General
School (Romania), but also I did intensive private tutoring for students preparing their
university entrance examination. After that, I have escaped in Turkey in September 1988
and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I
immigrated to United States. The degree of difficulties of the problems is from easy
and medium to hard. The solutions of the problems are at the end of each chapter.
One can navigate back and forth from the text of the problem to its solution using
bookmarks. The book is especially a didactical material for the mathematical students
and instructors.
The Author

5
Problems in Geometry (9th
grade)
1. The measure of a regular polygon’s interior angle is four times bigger than
the measure of its external angle. How many sides does the polygon have?
Solution to Problem 1
2. How many sides does a convex polygon have if all its external angles are
obtuse?
3. Show that in a convex quadrilateral the bisector of two consecutive angles
forms an angle whose measure is equal to half the sum of the measures of
the other two angles.
4. Show that the surface of a convex pentagon can be decomposed into two
quadrilateral surfaces.
5. What is the minimum number of quadrilateral surfaces in which a convex
polygon with 9, 10, 11 vertices can be decomposed?
6. If (𝐴𝐵𝐶)̂ ≡ (𝐴′ 𝐵′ 𝐶′)̂ , then ∃ bijective function 𝑓 = (𝐴𝐵𝐶)̂ → (𝐴′ 𝐵′ 𝐶′)̂ such
that for ∀ 2 points 𝑃, 𝑄 ∈ (𝐴𝐵𝐶)̂ , ‖𝑃𝑄‖ = ‖𝑓(𝑃)‖, ‖𝑓(𝑄)‖, and vice versa.

6
7. If ∆𝐴𝐵𝐶 ≡ ∆𝐴′
𝐵′
𝐶′
then ∃ bijective function 𝑓 = 𝐴𝐵𝐶 → 𝐴′
𝐵′
𝐶′
such that
(∀) 2 points 𝑃, 𝑄 ∈ 𝐴𝐵𝐶, ‖𝑃𝑄‖ = ‖𝑓(𝑃)‖, ‖𝑓(𝑄)‖, and vice versa.
8. Show that if ∆𝐴𝐵𝐶~∆𝐴′
𝐵′
𝐶′
, then [𝐴𝐵𝐶]~[𝐴′
𝐵′
𝐶′].
9. Show that any two rays are congruent sets. The same property for lines.
10. Show that two disks with the same radius are congruent sets.
11. If the function 𝑓: 𝑀 → 𝑀′
is isometric, then the inverse function 𝑓−1
: 𝑀 → 𝑀′
is as well isometric.
12. If the convex polygons 𝐿 = 𝑃1, 𝑃2, … , 𝑃𝑛 and 𝐿′
= 𝑃1
′
, 𝑃2
′
, … , 𝑃𝑛
′
have |𝑃𝑖, 𝑃𝑖+1| ≡
|𝑃𝑖
′
, 𝑃𝑖+1
′
| for 𝑖 = 1, 2, … , 𝑛 − 1, and 𝑃𝑖 𝑃𝑖+1 𝑃𝑖+2
̂ ≡ 𝑃𝑖
′
𝑃𝑖+1
′
𝑃𝑖+2
′̂ , (∀) 𝑖 = 1, 2, … , 𝑛 −
2, then 𝐿 ≡ 𝐿′
and [𝐿] ≡ [𝐿′].
13. Prove that the ratio of the perimeters of two similar polygons is equal to
their similarity ratio.
14. The parallelogram 𝐴𝐵𝐶𝐷 has ‖𝐴𝐵‖ = 6, ‖𝐴𝐶‖ = 7 and 𝑑(𝐴𝐶) = 2. Find
𝑑(𝐷, 𝐴𝐵).

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15. Of triangles 𝐴𝐵𝐶 with ‖𝐵𝐶‖ = 𝑎 and ‖𝐶𝐴‖ = 𝑏, 𝑎 and 𝑏 being given
numbers, find a triangle with maximum area.
16. Consider a square 𝐴𝐵𝐶𝐷 and points 𝐸, 𝐹, 𝐺, 𝐻, 𝐼, 𝐾, 𝐿, 𝑀 that divide each side
in three congruent segments. Show that 𝑃𝑄𝑅𝑆 is a square and its area is
equal to
2
9
𝜎[𝐴𝐵𝐶𝐷].
17. The diagonals of the trapezoid 𝐴𝐵𝐶𝐷 (𝐴𝐵||𝐷𝐶) cut at 𝑂.
a. Show that the triangles 𝐴𝑂𝐷 and 𝐵𝑂𝐶 have the same area;
b. The parallel through 𝑂 to 𝐴𝐵 cuts 𝐴𝐷 and 𝐵𝐶 in 𝑀 and 𝑁. Show that
||𝑀𝑂|| = ||𝑂𝑁||.
18. 𝐸 being the midpoint of the non-parallel side [𝐴𝐷] of the trapezoid 𝐴𝐵𝐶𝐷,
show that 𝜎[𝐴𝐵𝐶𝐷] = 2𝜎[𝐵𝐶𝐸].
19. There are given an angle (𝐵𝐴𝐶)̂ and a point 𝐷 inside the angle. A line
through 𝐷 cuts the sides of the angle in 𝑀 and 𝑁. Determine the line 𝑀𝑁
such that the area ∆𝐴𝑀𝑁 to be minimal.
20. Construct a point 𝑃 inside the triangle 𝐴𝐵𝐶, such that the triangles 𝑃𝐴𝐵,
𝑃𝐵𝐶, 𝑃𝐶𝐴 have equal areas.
21. Decompose a triangular surface in three surfaces with the same area by
parallels to one side of the triangle.

8
22. Solve the analogous problem for a trapezoid.
23. We extend the radii drawn to the peaks of an equilateral triangle inscribed
in a circle 𝐿(𝑂, 𝑟), until the intersection with the circle passing through the
peaks of a square circumscribed to the circle 𝐿(𝑂, 𝑟). Show that the points
thus obtained are the peaks of a triangle with the same area as the
hexagon inscribed in 𝐿(𝑂, 𝑟).
24. Prove the leg theorem with the help of areas.
25. Consider an equilateral ∆𝐴𝐵𝐶 with ‖𝐴𝐵‖ = 2𝑎. The area of the shaded
surface determined by circles 𝐿(𝐴, 𝑎), 𝐿(𝐵, 𝑎), 𝐿(𝐴, 3𝑎) is equal to the area of
the circle sector determined by the minor arc (𝐸𝐹)̂ of the circle 𝐿(𝐶, 𝑎).
26. Show that the area of the annulus between circles 𝐿(𝑂, 𝑟2) and 𝐿(𝑂, 𝑟2) is
equal to the area of a disk having as diameter the tangent segment to
circle 𝐿(𝑂, 𝑟1) with endpoints on the circle 𝐿(𝑂, 𝑟2).
27. Let [𝑂𝐴], [𝑂𝐵] two ⊥ radii of a circle centered at [𝑂]. Take the points 𝐶 and
𝐷 on the minor arc 𝐴𝐵𝐹̂ such that 𝐴𝐶̂ ≡𝐵𝐷̂ and let 𝐸, 𝐹 be the projections of
𝐶𝐷 onto 𝑂𝐵. Show that the area of the surface bounded by [𝐷𝐹], [𝐹𝐸[𝐸𝐶]]
and arc 𝐶𝐷̂ is equal to the area of the sector determined by arc 𝐶𝐷̂ of the
circle 𝐶(𝑂, ‖𝑂𝐴‖).

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28. Find the area of the regular octagon inscribed in a circle of radius 𝑟.
29. Using areas, show that the sum of the distances of a variable point inside
the equilateral triangle 𝐴𝐵𝐶 to its sides is constant.
30. Consider a given triangle 𝐴𝐵𝐶 and a variable point 𝑀 ∈ |𝐵𝐶|. Prove that
between the distances 𝑥 = 𝑑(𝑀, 𝐴𝐵) and 𝑦 = 𝑑(𝑀, 𝐴𝐶) is a relation of 𝑘𝑥 +
𝑙𝑦 = 1 type, where 𝑘 and 𝑙 are constant.
31. Let 𝑀 and 𝑁 be the midpoints of sides [𝐵𝐶] and [𝐴𝐷] of the convex
quadrilateral 𝐴𝐵𝐶𝐷 and {𝑃} = 𝐴𝑀 ∩ 𝐵𝑁 and {𝑄} = 𝐶𝑁 ∩ 𝑁𝐷. Prove that the
area of the quadrilateral 𝑃𝑀𝑄𝑁 is equal to the sum of the areas of triangles
𝐴𝐵𝑃 and 𝐶𝐷𝑄.
32. Construct a triangle having the same area as a given pentagon.
33. Construct a line that divides a convex quadrilateral surface in two parts
with equal areas.
34. In a square of side 𝑙, the middle of each side is connected with the ends of
the opposite side. Find the area of the interior convex octagon formed in
this way.

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35. The diagonal [𝐵𝐷] of parallelogram 𝐴𝐵𝐶𝐷 is divided by points 𝑀, 𝑁, in 3
segments. Prove that 𝐴𝑀𝐶𝑁 is a parallelogram and find the ratio between
𝜎[𝐴𝑀𝐶𝑁] and 𝜎[𝐴𝐵𝐶𝐷].
36. There are given the points 𝐴, 𝐵, 𝐶, 𝐷, such that 𝐴𝐵 ∩ 𝐶𝐷 = {𝑝}. Find the
locus of point 𝑀 such that 𝜎[𝐴𝐵𝑀] = 𝜎[𝐶𝐷𝑀].
37. Analogous problem for 𝐴𝐵||𝐶𝐷.
38. Let 𝐴𝐵𝐶𝐷 be a convex quadrilateral. Find the locus of point 𝑥1 inside 𝐴𝐵𝐶𝐷
such that 𝜎[𝐴𝐵𝑀] + 𝜎[𝐶𝐷𝑀] = 𝑘, 𝑘 – a constant. For which values of 𝑘 the
desired geometrical locus is not the empty set?

11
Solutions
Solution to Problem 1.
180 (𝑛 − 2)
𝑛
= 4
180
5
 𝑛 = 10
Let 𝑛 = 3
𝑥1, 𝑥2, 𝑥3∢ ext
⟹
𝑥1 > 900
𝑥2 > 900
𝑥3 > 900
} ⟹ 𝑥1 + 𝑥2 + 𝑥3 > 2700
, so 𝑛 = 3 is possible.
Let 𝑛 = 4
𝑥1, 𝑥2, 𝑥3, 𝑥4∢ ext
⟹
𝑥1 > 900
⋮
𝑥3 > 900
} ⟹ 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 > 3600
, so 𝑛 = 4 is impossible.
Therefore, 𝑛 = 3.

12
m(𝐴𝐸𝐵̂) =
m(𝐷̂) + m(𝐶̂)
2
m(𝐴̂) + m(𝐵̂) + m(𝐶̂) + m(𝐷̂) = 360°
m(𝐴̂) + m(𝐵̂)
2
= 180° −
m(𝐶̂) + m(𝐷̂)
2
m(𝐴𝐸𝐵̂) = 180° −
m(𝐴̂)
2
−
m(𝐵̂)
2
=
= 180° − 180° +
m(𝐶̂) + m(𝐷̂)
2
=
m(𝐶̂) + (𝐷̂)
2
Let 𝐸𝐷𝐶̂ ⇒ 𝐴, 𝐵 ∈ int. 𝐸𝐷𝐶̂. Let 𝑀 ∈ |𝐴𝐵| ⇒ 𝑀 ∈ int. 𝐸𝐷𝐶̂ ⇒ |𝐷𝑀 ⊂ int. 𝐸𝐷𝐶̂, |𝐸𝐴| ∩
|𝐷𝑀 = ∅ ⇒ 𝐷𝐸𝐴𝑀 quadrilateral. The same for 𝐷𝐶𝐵𝑀.
9 vertices; 10 vertices; 11 vertices;
4 quadrilaterals. 4 quadrilaterals. 5 quadrilaterals.

13
We assume that ABĈ ≡ A’B’C’̂ . We construct a function 𝑓: 𝐴𝐵𝐶̂ → 𝐴′𝐵′𝐶′̂ such that
{
𝑓 (B) = B′
if P ∈ |BA, 𝑓(P) ∈ B′A′
𝑃 ∈ |𝐵𝐶, 𝑓(𝑃) ∈ 𝐵′
𝐶′
such that ‖𝐵𝑃‖ = ‖𝐵′
𝑃′‖ where 𝑃′
= 𝑓(𝐹).
The so constructed function is bijective, since for different arguments there are
different corresponding values and ∀ point from 𝐴′𝐵′𝐶′ is the image of a single
point from 𝐴𝐵𝐶̂ (from the axiom of segment construction).
If 𝑃, 𝑄 ∈ this ray,
‖BP‖ = ‖B′
P′‖
‖BQ‖ = ‖B′Q′‖
} ⟹ ‖PQ‖ = ‖BQ‖ − ‖BP‖ = ‖B′Q′‖ − ‖B′
P′‖ = ‖P′Q′‖ = ‖𝑓(P), 𝑓(Q)‖.
If 𝑃, 𝑄 ∈ a different ray,
‖BP‖ = ‖B′
P′‖
‖BQ‖ = ‖B′Q′‖
PBQ̂ ≡ P′B′Q̂ ′
} ⟹ ∆PBQ = ∆P′
B′
Q′
‖PQ‖ = ‖P′
Q′‖ = ‖𝑓 (P), 𝑓 (Q)‖.
Vice versa.
Let 𝑓 ∶ ABC → A′
B′
C′
such that 𝑓 bijective and ‖PQ‖ = ‖𝑓(P), 𝑓(Q)‖.
Let 𝑃, 𝑄 ∈ |𝐵𝐴 and 𝑅𝑆 ∈ |𝐵𝐶.

14
‖PQ‖ = ‖P′
Q′‖
‖PS‖ = ‖P′
S′‖
‖QS‖ = ‖Q′S′‖
} ⟹ ∆PQS ≡ ∆P′
Q′
S′
⟹ QPŜ ≡ Q′P′S′̂ ⟹ BPŜ ≡ B′P′S′̂ (1);
‖𝑃𝑆‖ = ‖𝑃′𝑆′‖
‖𝑅𝑃‖ = ‖𝑅′𝑃′‖
‖𝑃𝑆‖ = ‖𝑃′𝑆′‖
} ⟹ ∆PRS ≡ ∆P′R′S′′
⟹ PSB̂ ≡ P′S′B′̂ (2).
From (1) and (2) ⟹ PBĈ ≡ P′B′S′̂ (as diff. at 180°) i.e. ABĈ ≡ A′B′C′̂ .
Let ∆𝐴𝐵𝐶 ≡ ∆𝐴′𝐵′𝐶′.
We construct a function 𝑓 ∶ 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ such that 𝑓(𝐴) = 𝐴′, 𝑓(𝐵) = 𝐵′, 𝑓(𝐶) = 𝐶′
and so 𝑃 ∈ |𝐴𝐵| → 𝑃′ = 𝑓(𝑃) ∈ |𝐴′𝐵′| such that ||𝐴𝑃|| = ||𝐴′𝑃′||;
𝑃 ∈ |𝐵𝐶| → 𝑃′ = 𝑓(𝑃) ∈ |𝐵′𝐶′| such that ||𝐵𝑃|| = ||𝐵′𝑃′||;
𝑃 ∈ |𝐶𝐴| → 𝑃′ = 𝑓(𝑃) ∈ |𝐶′𝐴′| such that ||𝐶𝑃|| = ||𝐶′𝑃′||.
The so constructed function is bijective.
Let 𝑃 ∈ |𝐴𝐵| and 𝑎 ∈ |𝐶𝐴| ⟹ 𝑃′ ∈ |𝐴′𝐵′| and 𝑄′ ∈ |𝐶′𝐴′|.
‖AP‖ = ‖A′P′‖
‖CQ‖ = ‖C′Q′‖
‖CA‖ = ‖C′A′‖
} ⟹ ‖AQ‖ = ‖A′Q′‖; A ≡ A′
⟹ ∆APQ ≡ ∆A′P′Q′ ⟹ ‖PQ‖ = ‖P′Q′‖.
Similar reasoning for (∀) point 𝑃 and 𝑄.
Vice versa.
We assume that ∃ a bijective function 𝑓 ∶ 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ with the stated
properties.
We denote 𝑓(A) = A′′
, 𝑓(B) = B′′
, 𝑓(C) = C′′
⟹ ‖AB‖ = ‖A′′
B′′‖, ‖BC‖ = ‖B′′
C′′‖, ‖AC‖ = ‖A′′
C′′‖∆ABC = ∆A′′
B′′
C′′
.
Because 𝑓(ABC) = 𝑓([AB] ∪ [BC] ∪ [CA]) = 𝑓([AB]) ∪ 𝑓([BC]) ∪ 𝑓([CA])
= [A′′B′′] ∪ [B′′
C′′][C′′
A′′] = A′′
B′′
C′′
.
But by the hypothesis 𝑓(𝐴𝐵𝐶) = 𝑓(𝐴’𝐵’𝐶’), therefore
A′′
B′′
C′′
= ∆A′
B′
C′
⟹ ∆ABC ≡ ∆A′
B′
C′
.

15
If ∆𝐴𝐵𝐶~∆𝐴’𝐵’𝐶’ then (∀) 𝑓: 𝐴𝐵𝐶 → 𝐴’𝐵’𝐶’ and 𝑘 > 0 such that:
||𝑃𝑄|| = 𝑘 ||𝑓(𝑃), 𝑓(𝑄)||, 𝑃, 𝑄 ∈ 𝐴𝐵𝐶;
∆𝐴𝐵𝐶~∆𝐴′𝐵′𝐶′ ⟹
‖AB‖
‖A′B′‖
=
‖BC‖
‖B′C′‖
=
‖CA‖
‖C′A′‖
= 𝑘
Â ≡ A′̂ ; B̂ ≡ B′̂ ; Ĉ ≡ C′̂
} ⟹
‖AB‖ = 𝑘‖A′B′‖
‖BC‖ = 𝑘‖B′C′‖
‖CA‖ = 𝑘‖C′A′‖
.
We construct a function 𝑓: 𝐴𝐵𝐶 → 𝐴′𝐵′𝐶′ such that 𝑓(𝐴) = 𝐴′
, 𝑓(𝐵) = 𝐵′
, 𝑓(𝐶) = 𝐶′;
if 𝑃 ∈ |𝐵𝐶| → 𝑃 ∈ |𝐵′𝐶′| such that ||𝐵𝑃|| = 𝑘||𝐵′𝑃′||;
if 𝑃 ∈ |𝐶𝐴| → 𝑃 ∈ |𝐶′𝐴′| such that ||𝐶𝑃|| = 𝑘||𝐶′𝑃′||; 𝑘 – similarity constant.
Let 𝑃, 𝑄 ∈ 𝐴𝐵 such that 𝑃 ∈ |𝐵𝐶|, 𝑄 ∈ |𝐴𝐶| ⟹ 𝑃′ ∈ |𝐵′𝐶′| and ||𝐵𝑃|| = 𝑘||𝐵′𝑃′||
Q′
∈ |A′
C′| and ‖CQ‖ = 𝑘‖C′Q′‖ (1);
As ‖BC‖ = 𝑘‖B′
C′‖ ⟹ ‖PC‖ = ‖BC‖ − ‖BP‖ = 𝑘‖B′
C′‖ − 𝑘‖B′
P′‖ =
= 𝑘(‖B′C′‖ − ‖B′P′‖) = 𝑘‖P′C′‖ (2);
Ĉ ≡ C′̂ (3).
From (1), (2), and (3) ⟹ ∆PCQ~∆P′
C′
Q′
⟹ ‖PQ‖ = 𝑘‖P′Q′‖ .
Similar reasoning for 𝑃, 𝑄 ∈ 𝐴𝐵𝐶.
We also extend the bijective function previously constructed to the interiors of
the two triangles in the following way:
Let 𝑃 ∈ int. 𝐴𝐵𝐶 and we construct 𝑃’ ∈ int. 𝐴’𝐵’𝐶’ such that ||𝐴𝑃|| = 𝑘||𝐴’𝑃’|| (1).
Let 𝑄 ∈ int. 𝐴𝐵𝐶 → 𝑄′ ∈ int. 𝐴’𝐵’𝐶’ such that BAQ̂ ≡ B’A’Q’̂ and ||𝐴𝑄|| = 𝑘||𝐴’𝑄’|| (2).

16
From (1) and (2),
AP
A′P′
=
AQ
A′Q′
= 𝑘, PAQ̂ ≡ P′A′Q′̂ ⟹ ∆APQ ~ ∆A′
P′
Q′
⟹ ‖PQ‖ = 𝑘‖P′
Q′‖,
but 𝑃, 𝑄 ∈ [𝐴𝐵𝐶], so [𝐴𝐵𝐶] ~[𝐴′𝐵′𝐶′].
a. Let |𝑂𝐴 and |𝑂′𝐴′ be two rays:
Let 𝑓: |𝑂𝐴 → |𝑂′𝐴′ such that 𝑓(𝑂) = 𝑂′ and 𝑓(𝑃) = 𝑃′ with ||𝑂𝑃|| = ||𝑂′𝑃′|.
The so constructed point 𝑃′ is unique and so if 𝑃 ≠ 𝑄 ⟹ ||𝑂𝑃|| ≠ ||𝑂𝑄|| ⟹
||𝑂′𝑃′|| ≠ ||𝑂′𝑄′|| ⟹ 𝑃′ ≠ 𝑄′ and (∀)𝑃′ ∈ |𝑂′𝐴′ (∃) a single point 𝑃 ∈ |𝑂𝐴 such that
||𝑂𝑃|| = ||𝑂′𝑃′||.
The constructed function is bijective.
If 𝑃, 𝑄 ∈ |𝑂𝐴, 𝑃 ∈ |𝑂𝑄| → 𝑃′
𝑄′
∈ |𝑂′
𝐴′
such that ‖OP‖ = ‖O′P′‖;‖OQ‖ = ‖O′
Q′‖ ⟹
‖PQ‖ = ‖OQ‖ − ‖OP‖ = ‖O′
Q′‖ − ‖O′
P′‖ = ‖P′
Q′‖(∀)P; 𝑄 ∈ |OA
⟹ the two rays are congruent.
b. Let 𝑑 and 𝑑′ be two lines.
Let 𝑂 ∈ 𝑑 and 𝑂′ ∈ 𝑑′. We construct a function 𝑓: 𝑑 → 𝑑′ such that 𝑓(𝑂) = 𝑂′ and
𝑓 (|𝑂𝐴) = |𝑂′𝐴′ and 𝑓 (|𝑂𝐵) = |𝑂′𝐵′ as at the previous point.
It is proved in the same way that 𝑓 is bijective and that ||𝑃𝑄|| = ||𝑃′𝑄′|| when 𝑃
and 𝑄 belong to the same ray.

17
If 𝑃, 𝑄 belong to different rays:
‖𝑂𝑃‖ = ‖𝑂′𝑃′‖
‖𝑂𝑄‖ = ‖𝑂′𝑄′‖
} ⟹ ‖𝑃𝑄‖ = ‖𝑂𝑃‖ + ‖𝑂𝑄‖ = ‖𝑂′
𝑃′‖ + ‖𝑂′
𝑄′‖ = ‖𝑃′𝑄′‖
and so the two rays are congruent.
We construct a function 𝑓: 𝐷 → 𝐷′ such that 𝑓(𝑂) = 𝑂′, 𝑓(𝐴) = 𝐴′ and a point
(∀) 𝑃 ∈ 𝐷 → 𝑃′ ∈ 𝐷′ which are considered to be positive.
From the axiom of segment and angle construction ⟹ that the so constructed
function is bijective, establishing a biunivocal correspondence between the elements
of the two sets.
Let 𝑄 ∈ 𝐷 → 𝑄′ ∈ 𝐷′ such that ||𝑂𝑄′|| = ||𝑂𝑄||; 𝐴𝑂𝑄̂ ≡ 𝐴′ 𝑂′ 𝑄′̂ .
As:
‖𝑂𝑃‖ = ‖𝑂′𝑃′‖
‖𝑂𝑄‖ = ‖𝑂′𝑄′‖
𝑃𝑂𝑄̂ ≡ 𝑃′ 𝑂′ 𝑄′̂
} ⟹ ∆𝑂𝑃𝑄 ≡ 𝑃′ 𝑂′ 𝑄′̂ ⟹ ‖𝑃𝑄‖ = ‖𝑃′
𝑄′‖, (∀) 𝑃, 𝑄 ∈ 𝐷 ⟹ 𝐷 ≡ 𝐷′
.

18
𝑓: 𝑀 → 𝑀′ is an isometry ⟹ 𝑓 is bijective and (∀) 𝑃, 𝑄 ∈ 𝑀 we have ||𝑃𝑄|| =
||𝑓(𝑃), 𝑓(𝑄)||, 𝑓 – bijective ⟹ 𝑓 – invertible and 𝑓−1
– bijective.
‖𝑃′𝑄′‖ = ‖𝑓(𝑃); 𝑓(𝑄)‖ = ‖𝑃𝑄‖
‖𝑓−1(𝑃′); 𝑓−1(𝑄′)‖ = ‖𝑓−1
(𝑓(𝑃)), 𝑓−1
(𝑓(𝑄))‖ = ‖𝑃𝑄‖
} ⟹
‖𝑃′𝑄′‖ = ‖𝑓−1
(𝑓(𝑃′)), 𝑓−1
(𝑓(𝑄′))‖, (∀)𝑃′
, 𝑄′
∈ 𝑀,
therefore 𝑓−1
: 𝑀′ → 𝑀 is an isometry.
We construct a function 𝑓 such that 𝑓(𝑃𝑖) = 𝑃𝑖
′
, 𝑖 = 1, 2, … , 𝑛, and if 𝑃 ∈
| 𝑃𝑖, 𝑃𝑖+1|.
The previously constructed function is also extended inside the polygon as
follows: Let 𝑂 ∈ int. 𝐿 → 𝑂′ ∈ int. 𝐿′ such that 𝑂𝑃𝑖 𝑃𝑖+1
̂ ≡ 𝑂′𝑃′𝑖 𝑃′𝑖+1
̂ and ‖𝑂𝑃𝑖‖ =
‖𝑂′𝑃′𝑖‖. We connect these points with the vertices of the polygon. It can be easily
proved that the triangles thus obtained are congruent.
We construct the function 𝑔: [𝐿] → [𝐿′] such that
𝑔(𝑃) = {
𝑓(𝑃), if 𝑃 ∈ 𝐿
𝑂′
, if 𝑃 = 𝑂
𝑃′
, if 𝑃 ∈ [𝑃𝑖 𝑂𝑃𝑖+1] such that
𝑃𝑖 𝑂𝑃̂ ≡ 𝑃𝑖
′ 𝑂′ 𝑃′̂ (∀)𝑖 = 1, 2, … , 𝑛 − 1
The so constructed function is bijective (∀) 𝑃, 𝑄 ∈ [𝐿]. It can be proved by the
congruence of the triangles 𝑃𝑂𝑄 and 𝑃′𝑂′𝑄′ that ||𝑃𝑄|| = ||𝑃′𝑄′||, so [𝐿] = [𝐿′]
⟹ if two convex polygons are decomposed
in the same number of triangles
respectively congruent,
they are congruent.

19
𝐿 = 𝑃1 𝑃2 … , 𝑃𝑛; 𝐿′
= 𝑃1
′
𝑃2
′
… , 𝑃𝑛
′
𝐿~𝐿′ ⟹ (∃)𝐾 > 0 and 𝑓: 𝐿 → 𝐿′ such that ‖𝑃𝑄‖ = 𝑘‖𝑓(𝑃)𝑓(𝑄)‖ (∀)𝑃, 𝑄 ∈ 𝐿,
and 𝑃𝐼
′
= 𝑓(𝑃𝑖).
Taking consecutively the peaks in the role of 𝑃 and 𝑄, we obtain:
‖𝑃1 𝑃2‖ = 𝑘‖𝑃1
′
𝑃2
′‖ ⇒
‖𝑃1 𝑃2‖
‖𝑃1
′
𝑃2
′‖
= 𝑘
‖𝑃2 𝑃3‖ = 𝑘‖𝑃2
′
𝑃3
′‖ ⇒
‖𝑃2 𝑃3‖
‖𝑃2
′
𝑃3
′‖
= 𝑘
⋮
‖𝑃𝑛−1 𝑃𝑛‖ = 𝑘‖𝑃𝑛−1
′
𝑃𝑛
′‖ ⇒
‖𝑃𝑛−1 𝑃𝑛‖
‖𝑃𝑛−1
′
𝑃𝑛
′‖
= 𝑘
‖𝑃𝑛 𝑃1‖ = 𝑘‖𝑃𝑛
′
𝑃1
′‖ ⇒
‖𝑃𝑛 𝑃1‖
‖𝑃𝑛
′ 𝑃1
′‖
= 𝑘
}
⟹
⟹ 𝑘 =
‖𝑃1 𝑃2‖
‖𝑃1
′
𝑃2
′‖
=
‖𝑃2 𝑃3‖
‖𝑃2
′
𝑃3
′‖
= ⋯ =
‖𝑃1 𝑃2‖ + ‖𝑃2 𝑃3‖ + ⋯ + ‖𝑃𝑛−1 𝑃𝑛‖ + ‖𝑃𝑛 𝑃1‖
‖𝑃1
′
𝑃2
′‖ + ‖𝑃2
′
𝑃3
′‖ + ⋯ + ‖𝑃𝑛−1
′
𝑃𝑛
′‖ + ‖𝑃𝑛
′ 𝑃1
′‖
=
𝑃
𝑃′
.
𝜎[𝐴𝐷𝐶] =
2∙7
2
= 7; 𝜎[𝐴𝐵𝐶𝐷] = 2 ∙ 7 = 14 = 6‖𝐷𝐹‖ ⟹ ‖𝐷𝐹‖ =
14
6
=
7
3
.

20
ℎ = 𝑏 ∙ sin 𝐶 ≤ 𝑏;
𝜎[𝐴𝐵𝐶] =
𝑎∙ℎ
2
is max. when ℎ is max.
max. ℎ = 𝑏 when sin𝐶 = 1
⇒ 𝑚(𝐶) = 90 ⇒ 𝐴𝐵𝐶 has a right angle at 𝐶.
‖𝑀𝐷‖ = ‖𝐷𝐼‖ ⟹ 𝑀𝐷𝐼 – an isosceles triangle.
⟹ 𝑚(𝐷𝑀𝐼̂) = 𝑚(𝑀𝐼𝐷̂) = 450
;
The same way, 𝑚(𝐹𝐿𝐴̂ ) = 𝑚(𝐴𝐹𝐿̂ ) = 𝑚(𝐵𝐸𝐻̂) = 𝑚(𝐸𝐻𝐵̂).
‖𝑅𝐾‖ ⟹ ‖𝑆𝑃‖ = ‖𝑃𝑄‖ = ‖𝑄𝑅‖ = ‖𝑅𝑆‖ ⟹ 𝑆𝑅𝑄𝑃 is a square.
‖𝐴𝐵‖ = 𝑎, ‖𝐴𝐸‖ =
2𝑎
3
, ‖𝑀𝐼‖ = √
4𝑎2
9
+
4𝑎2
9
=
2𝑎√2
3
;
2‖𝑅𝐼‖2
=
𝑎2
9
⟹ ‖𝑅𝐼‖2
=
𝑎2
18
⟹ ‖𝑅𝐼‖ =
𝑎
3√2
=
𝑎√2
6
;
‖𝑆𝑅‖ =
2𝑎√2
3
− 2
𝑎√2
6
=
𝑎√2
3
;
𝜎[𝑆𝑅𝑄𝑃] =
2𝑎2
9
=
2
9
𝜎[𝐴𝐵𝐶𝐷].
𝜎[𝐴𝐶𝐷] =
‖𝐷𝐶‖ ∙ ‖𝐴𝐸‖
2
𝜎[𝐵𝐶𝐷] =
‖𝐷𝐶‖ ∙ ‖𝐵𝐹‖
2
‖𝐴𝐸‖ = ‖𝐵𝐹‖ }
⟹ 𝜎[𝐴𝐶𝐷] = 𝜎[𝐵𝐶𝐷]

21
𝜎[𝐴𝑂𝐷] = 𝜎[𝐴𝑀𝑂] + 𝜎[𝑀𝑂𝐷]
𝜎[𝐴𝑀𝑂] = 𝜎[𝑀𝑃𝑂] =
‖𝑀𝑂‖ ∙ ‖𝑂𝑃‖
2
𝜎[𝑀𝑂𝐷] = 𝜎[𝑀𝑂𝑄] =
‖𝑂𝑀‖ ∙ ‖𝑂𝑄‖
2 }
⟹ 𝜎[𝐴𝑂𝐷] =
‖𝑂𝑀‖(‖𝑂𝑃‖ + ‖𝑂𝑄‖)
2
=
‖𝑂𝑀‖ ∙ ℎ
2
The same way,
𝜎[𝐵𝑂𝐶] =
‖𝑂𝑁‖ ∙ ℎ
2
.
Therefore,
𝜎[𝐴𝑂𝐷] = 𝜎[𝐵𝑂𝐶] ⟹
‖𝑂𝑀‖ ∙ ℎ
2
=
‖𝑂𝑁‖ ∙ ℎ
2
⟹ ‖𝑂𝑀‖ = ‖𝑂𝑁‖.
‖𝐴𝐸‖ = ‖𝐸𝐷‖ ;
We draw 𝑀𝑁 ⊥ 𝐴𝐵; 𝐷𝐶;
‖𝐸𝑁‖ = ‖𝐸𝑀‖ =
ℎ
2
;
𝜎[𝐵𝐸𝐶] =
(‖𝐴𝐵‖ + ‖𝐷𝐶‖) ∙ ℎ
2
−
‖𝐴𝐵‖ ∙ ℎ
4
−
‖𝐷𝐶‖ ∙ ℎ
4
=
(‖𝐴𝐵‖ + ‖𝐷𝐶‖) ∙ ℎ
4
=
1
2
𝜎[𝐴𝐵𝐶𝐷];
Therefore, [𝐴𝐵𝐶𝐷] = 2𝜎[𝐵𝐸𝐶] .
𝜎[𝐴𝐸𝐷𝑁′] is ct. because 𝐴, 𝐸, 𝐷, 𝑁′ are fixed points.
Let a line through 𝐷, and we draw ∥ to sides 𝑁𝐷 and 𝐷𝐸.
No matter how we draw a line through 𝐷, 𝜎[𝑄𝑃𝐴] is formed of: 𝜎[𝐴𝐸𝐷𝑁] +
𝜎[𝑁𝑃𝑂] + 𝜎[𝐷𝐸𝑄].
We have 𝜎[𝐴𝐸𝐷𝑁] constant in all triangles 𝑃𝐴𝑄.

22
Let’s analyse:
𝜎[𝑃𝑁′
𝐷] + 𝜎[𝐷𝐸𝑄] =
‖𝑁′𝐷‖ ∙ ℎ1
2
+
‖𝐸𝑄′‖ ∙ ℎ2
2
=
‖𝑁′𝐷‖
2
(ℎ1 +
‖𝐸𝑄‖
‖𝑁𝐷‖
∙ ℎ2)
=
‖𝑁′𝐷‖
2
∙ (ℎ1 +
ℎ2
ℎ1
∙ ℎ2) =
‖𝑁′
𝐷‖
2ℎ1
[(ℎ1 − ℎ2)2
+ 2ℎ1ℎ2].
∆𝐴𝑀𝑁 is minimal when ℎ1 = ℎ2 ⟹ 𝐷 is in the middle of |𝑃𝑄|. The construction is
thus: ∆𝐴𝑁𝑀 where 𝑁𝑀 || 𝐸𝑁′. In this case we have |𝑁𝐷| ≡ |𝐷𝑀|.
‖𝐵𝐶‖ ∙ ‖𝐴𝐴′‖
2
Let the median be |𝐴𝐸|, and 𝑃 be the centroid of the triangle.
Let 𝑃𝐷 ⊥ 𝐵𝐶. 𝜎[𝐵𝑃𝐶] =
‖𝐵𝐶‖∙‖𝑃𝐷‖
2
.
𝐴𝐴′ ⊥ 𝐵𝐶
𝑃𝐷 ⊥ 𝐵𝐶
} ⇒ 𝐴𝐴′
∥ 𝑃𝐷 ⇒ ∆𝑃𝐷𝐸~∆𝐴𝐴′
𝐸 ⇒
‖𝑃𝐷‖
‖𝐴𝐴′‖
=
‖𝑃𝐸‖
‖𝐴𝐸‖
=
1
3
⇒ ‖𝑃𝐷‖ =
‖𝐴𝐴′‖
3
⇒ 𝜎[𝐵𝑃𝐶]
=
‖𝐵𝐶‖ ∙
‖𝐴𝐴′‖
3
2
=
1
3
‖𝐵𝐶‖ ∙ ‖𝐴𝐴′‖
2
=
1
3
𝜎[𝐴𝐵𝐶].
We prove in the same way that 𝜎[𝑃𝐴𝐶] = 𝜎[𝑃𝐴𝐵] =
1
3
𝜎[𝐴𝐵𝐶], so the specific
point is the centroid.

23
Let 𝑀, 𝑁 ∈ 𝐴𝐵 such that 𝑀 ∈ |𝐴𝑁|.
We take 𝑀𝑀′ ∥ 𝐵𝐶, 𝑀𝑁′ ∥ 𝐵𝐶.
∆𝐴𝑀𝑀′
~∆𝐴𝐵𝐶 ⇒
𝜎[𝐴𝑀𝑀′]
𝜎[𝐴𝐵𝐶]
= (
𝐴𝑀
𝐴𝐵
)
𝜎[𝐴𝑀𝑀′] =
1
3
𝜎, (
‖𝐴𝑀‖
‖𝐴𝐵‖
)
2
=
1
3
, ‖𝐴𝑀‖ =
‖𝐴𝐵‖
√3
; ∆𝐴𝑁𝑁′~∆𝐴𝐵𝐶 ⟹
𝜎[𝐴𝑁𝑁′]
𝜎[𝐴𝐵𝐶]
= (
‖𝐴𝑁‖
‖𝐴𝐵‖
)
2
𝜎[𝐴𝑁𝑁′] =
2
3
𝜎[𝐴𝐵𝐶]
}
⟹ (
‖𝐴𝑁‖
‖𝐴𝐵‖
)
2
=
2
3
⟹ ‖𝐴𝑁‖ = √
2
3
‖𝐴𝐵‖ .
‖𝑂𝐷‖ = 𝑎, ‖𝑂𝐴‖ = 𝑏 ;
𝜎[∆𝐶𝑀′𝑀] = 𝜎[𝑀𝑀′𝑁′𝑁] = 𝜎[𝑁𝑁′𝐵𝐴] =
1
3
𝜎[𝐴𝐵𝐶𝐷] ;
∆𝑂𝐷𝐶~∆𝑂𝐴𝐵 ⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝑂𝐴𝐵]
=
‖𝑂𝐷‖2
‖𝑂𝐴‖2
=
𝑎
𝑏
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝑂𝐴𝐵] − 𝜎[𝑂𝐷𝐶]
=
𝑎2
𝑏2 − 𝑎2
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝐴𝐵𝐶𝐷]
=
𝑎2
𝑏2 − 𝑎2
(1)

24
∆𝑂𝐷𝐶~∆𝑂𝑀𝑀′
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝑂𝑀𝑀′]
= (
‖𝑂𝐷‖
‖𝑂𝑀‖
)
2
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝑂𝑀𝑀′] − 𝜎[𝑂𝐷𝐶]
=
𝑎2
‖𝑂𝑀‖2
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝐷𝐶𝑀𝑀′]
=
𝑎2
‖𝑂𝑀‖2 − 𝑎2
⟹
𝜎[𝑂𝐷𝐶]
1
3 𝜎[𝐴𝐵𝐶𝐷]
=
𝑎2
‖𝑂𝑀‖2 − 𝑎2
(2)
∆𝑂𝑁𝑁′
~∆𝑂𝐷𝐶 ⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝐷𝑁𝑁′]
=
‖𝑂𝐷‖
‖𝑂𝑁‖
=
𝑎2
‖𝑂𝑁‖
⟹
𝜎[𝐼𝐷𝐶]
𝜎[𝑂𝑁𝑁′] − 𝜎[𝑂𝐷𝐶]
=
𝑎2
‖𝑂𝑁‖2
⟹
𝜎[𝑂𝐷𝐶]
𝜎[𝐷𝐶𝑁𝑁′]
=
𝑎2
‖𝑂𝑁‖2 − 𝑎2
⟹
𝜎[𝑂𝐷𝐶]
2
3 𝜎[𝐴𝐵𝐶𝐷]
=
𝑎2
‖𝑂𝑁‖2 − 𝑎2
(3)
We divide (1) by (3):
2
3
=
‖𝑂𝑁‖2
− 𝑎2
𝑏2 − 𝑎2
⟹ 3‖𝑂𝑁‖2
− 3𝑎2
= 2𝑏2
− 2𝑎2
⟹ ‖𝑂𝑁‖2
=
𝑎2
+ 2𝑏2
3
.
‖𝑂𝐴‖ = 𝑟 → ‖𝐷𝐸‖ = 2𝑟; 𝜎hexagon =
3𝑟2
√2
2
(1)
𝐷𝐸𝐹𝑂 a square inscribed in the circle with radius 𝑅 ⟹
⟹ 𝑙4 = 𝑅√2 = ‖𝐷𝐸‖ ⟹ 𝑃√2 = 2𝑟 ⟹ 𝑅 = 𝑟√2
‖𝑂𝑀‖ = 𝑅 = 𝑟√2
𝜎[𝑂𝑀𝑁] =
‖𝑂𝑀‖ ∙ ‖𝑂𝑁‖ sin 120
2
=
𝑟√2 ∙ 𝑟√2 ∙
√3
2
2
=
𝑟2
√3
2
𝜎[𝑀𝑁𝑃] = 3𝜎[𝑂𝑀𝑁] = 3
𝑟2
√3
2
=
3𝑟2
√2
2
(2)
From (1) and (2) ⟹ 𝜎[𝑀𝑁𝑃] = 𝜎hexagon.

25
‖𝐴𝐵‖2
= ‖𝐵𝐶‖ ∙ ‖𝐵𝐴′‖
We construct the squares 𝐵𝐶𝐸𝐷 on the hypotenuse and 𝐴𝐵𝐹𝐺 on the leg.
We draw 𝐴𝐴′ ⊥ 𝐵𝐶.
𝜎[𝐴𝐵𝐹𝐺] = ‖𝐴𝐵‖2
𝜎[𝐴′𝐵𝐷𝐻] = ‖𝐵𝐷‖ ∙ ‖𝐵𝐴′‖ = ‖𝐵𝐶‖ ∙ ‖𝐵𝐴′‖ …
𝜎(𝑠1) = 𝜎[𝐴𝐵𝐶] − 3𝜎[sect. 𝐴𝐷𝐻]
𝑙2
√3
4
=
(2𝑎)2
√3
4
= 𝑎2
√3
𝜎[sect. 𝐴𝐷𝐻] =
𝑟2
2
𝑚(𝐷𝐻̂ )
𝑚(𝐷𝐻̂ ) =
𝜋
180
𝑚(𝐷𝐻̂ ) =
𝜋
180
∙ 600
=
𝜋
3
𝜎[sect. 𝐴𝐷𝐻] =
𝑎2
2
∙
𝜋
3
=
𝜋𝑎2
2
𝜎(𝑠1) = 𝑎2
√3 − 3
𝜋𝑎2
6
= 𝑎2
√3 −
𝜋𝑎2
2
(1)

26
𝜎(𝑠2) = 𝜎[sect. 𝐴𝐸𝐺] − 𝜎[sect. 𝐴𝐵𝐶] − 𝜎[sect. 𝐸𝐶𝐹] − 𝜎[sect. 𝐺𝐵𝐹]
=
(3𝑎)2
2
∙
𝜋
180
∙ 60 − 𝑎2
√3 −
𝑎2
2
∙
𝜋
180
∙ 120 =
9𝑎2
2
∙
𝜋
3
− 𝑎2
√3 −
𝜋𝑎2
3
−
𝜋𝑎2
3
=
3𝜋𝑎2
2
−
2𝜋𝑎2
3
− 𝑎2
√3 (2)
From (1) and (2)
⟹ 𝜎(𝑠1) + 𝜎(𝑠2) =
3𝜋𝑎2
2
−
2𝜋𝑎2
3
−
𝜋𝑎2
2
=
2𝜋𝑎2
6
=
𝜋𝑎2
3
.
‖𝐴𝐷‖2
= 𝑟2
2
− 𝑟1
2
𝜎[𝐿(𝑂, 𝑟1)] = 𝜋𝑟1
2
𝜎[𝐿(𝑂, 𝑟2)] = 𝜋𝑟2
2
𝜎[annulus] = 𝜋𝑟2
2
− 𝜋𝑟1
2
= 𝜋(𝑟2
2
− 𝑟1
2) (1)
𝜎[disk diameter‖𝐴𝐵‖] = 𝜋‖𝐴𝐷‖2
= 𝜋(𝑟2
2
− 𝑟1
2) (2)
From (1) and (2) ⟹ 𝜎[annulus] = 𝜎[disk diameter].

27
𝜎[𝐶𝐷𝐸𝐹] =
𝜎[𝐶𝐷𝐷′𝐶′]
2
𝜎[𝐶𝐷𝐷′𝐶′] = 𝜎seg[𝐶𝐷𝐵𝐷′𝐶′] − 𝜎seg[𝐷𝐵𝐷′]
We denote 𝑚(𝐴𝐶̂ ) = 𝑚(𝐵𝐷̂ ) = 𝑎 ⟹ 𝑚(𝐶𝐷̂ ) =
𝜋
2
− 2𝑎
𝜎sect. =
𝑟2
2
(𝛼 − sin 𝛼)
𝜎[𝐶𝐷𝐵𝐷′𝐶′] =
𝑟2
2
[𝜋 − 2𝛼 − sin(𝜋 − 2𝛼)] = 𝑞
𝑟2
2
[𝜋 − 2𝛼 − sin(𝜋 − 2𝛼)]
𝜎𝐴 =
𝑟2
2
(2𝛼 − sin 2𝛼)
𝜎[𝐶𝐷𝐷′𝐶′] = 𝜎[𝐶𝐷𝐵𝐷′𝐶′] − 𝜎[𝐷𝐵𝐷′] =
𝑟2
2
(𝜋 − 2𝛼 − sin2𝛼) =
𝑟2
2
(2𝛼 − sin2𝛼)
=
𝑟2
2
(𝜋 − 2𝛼 − sin 2𝛼 − 2𝛼 + sin 2𝛼) =
𝑟2
2
(𝜋 − 4𝛼)
⟹ 𝜎[𝐶𝐷𝐸𝐹] =
𝜎[𝐶𝐷𝐷′
𝐶′]
2
=
𝑟2
4
(𝜋 − 4𝛼) =
𝑟2
2
(
𝜋
2
− 2𝛼) (1)
𝜎[sect. 𝐶𝑂𝐷] =
𝑟2
2
𝑚(𝐶𝐷̂ ) =
𝑟2
2
(
𝜋
2
− 2𝛼) (2)
From (1) and (2) ⟹ 𝜎[𝐶𝐷𝐸𝐹] = 𝜎[sect. 𝐶𝑂𝐷].
‖𝑂1 𝐹‖ = ‖𝑂𝐸‖
𝜎[square] = ‖𝐷𝐸‖2
= ‖𝑂𝐴‖2
=
𝑏𝑐
2
= 𝑉[𝐴𝐵𝐶]
𝜇(𝐴𝑂𝐵̂) =
𝜋
4
𝜎[𝐴𝑂𝐵] =
𝑟2
sin
𝜋
4
2
=
𝑟2 √2
2
2
=
𝑟2
√2
4
𝜎[orthogon] = 8 ∙
𝑟2
√2
4
= 2√2𝑟2

28
𝜎[𝐴𝐵𝐶] = 𝜎[𝐴𝑀𝐵] + 𝜎[𝐴𝑀𝐶] + 𝜎[𝑀𝐵𝐶]
⟹ 𝑎ℎ 𝑎 = 𝑎𝑑3 + 𝑎𝑑2 + 𝑎𝑑1
𝑑1 + 𝑑2 + 𝑑3 = ℎ 𝑎 (𝑎 is the side of equilateral triangle)
⟹ 𝑑1 + 𝑑2 + 𝑑3 =
𝑎√3
2
(because ℎ 𝑎 =
𝑎√3
2
).
𝐴𝐵𝐶 − given ∆ ⟹ 𝑎, 𝑏, 𝑐, ℎ − constant
𝑎ℎ
2
𝜎[𝐴𝐵𝐶] = 𝜎[𝐴𝑀𝐵] + 𝜎[𝐴𝑀𝐶]
⟹
𝑎ℎ
2
=
𝑐𝑥
2
+
𝑏𝑦
2
⟹ 𝑐𝑥 + 𝑏𝑦 = 𝑎𝑏 ⟹
𝑐
𝑎ℎ
𝑥 +
𝑏
𝑎ℎ
𝑦 = 1 ⟹ 𝑘𝑥 + 𝑙𝑦 = 1,
where 𝑘 =
𝑐
𝑎ℎ
and 𝑙 =
𝑏
𝑎ℎ
.

29
We draw 𝐴𝐴′
⊥ 𝐵𝐶; 𝑁𝑁′
⊥ 𝐵𝐶; 𝐷𝐷′ ⊥ 𝐵𝐶 ⇒
𝐴𝐴′
∥ 𝑁𝑁′
∥ 𝐷𝐷′
‖𝐴𝑁‖ = ‖𝑁𝐷‖
} ⟹ 𝑀𝑁′
median line in
the trapezoid 𝐴𝐴′
𝐷′
𝐷 ⟹ ‖𝑁𝑁′‖ =
‖𝐴𝐴′‖+‖𝐷𝐷′‖
2
, 𝜎[𝐵𝐶𝑁] =
‖𝐵𝐶‖+‖𝑁𝑁′‖
2
.
First, we construct a quadrilateral with the same area as the given pentagon. We
draw through C a parallel to BD and extend |AB| until it intersects the parallel at M.
𝜎[𝐴𝐵𝐶𝐷𝐸] = 𝜎[𝐴𝐵𝐷𝐸] + 𝜎[𝐵𝐶𝐷],
𝜎[𝐵𝐶𝐷] = 𝜎[𝐵𝐷𝑀] (have the vertices on a parallel at the base).
Therefore, 𝜎[𝐴𝐵𝐶𝐷𝐸] = 𝜎[𝐴𝑀𝐷𝐸].
Then, we consider a triangle with the same area as the quadrilateral 𝐴𝑀𝐷𝐸.
We draw a parallel to 𝐴𝐷, 𝑁 is an element of the intersection with the same
parallel.
𝜎[𝐴𝑀𝐷𝐸] = 𝜎[𝐴𝐷𝐸] + 𝜎[𝐴𝐷𝐸] = 𝜎[𝐴𝐷𝐸] + 𝜎[𝐴𝐷𝑁] = 𝜎[𝐸𝐷𝑁].

30
‖𝐴𝐸‖ = ‖𝐸𝐶‖
‖𝐸𝐹‖𝐵𝐷 ⟹ 𝜎[𝐵𝐷𝐹] = 𝜎[𝐵𝐷𝐸]
𝜎[𝐴𝐵𝐹𝐷] = 𝜎[𝐴𝐵𝐸𝐷] (1)
𝜎[𝐴𝐷𝐸] = 𝜎[𝐷𝐸𝐶] equal bases and the same height;
𝜎[𝐴𝐷𝐸] = 𝜎[𝐷𝐸𝐶]
𝜎[𝐴𝐵𝐸𝐷] = 𝜎[𝐵𝐸𝐷𝐶] (2)
𝜎[𝐷𝐸𝐹] = 𝜎[𝐵𝐸𝐹] the same base and the vertices on parallel lines at the base;

31
It is proved in the same way that:
⟹ 𝑀𝑁𝑃𝑄 rhombus with right angle⟹ 𝑀𝑁𝑃𝑄 is a square.
It is proved in the same way that all the peaks of the octagon are elements of the
axis of symmetry of the square, thus the octagon is regular.

32
Consider the square separately.
‖𝑂𝑀‖ = ‖𝑁𝑀‖ = ‖𝑁𝐵‖
‖𝐷𝐶‖? ⟹ ∆𝑀𝑂𝐶 = ∆𝑁𝐵𝐴 ⟹ ‖𝑀𝐶‖ = ‖𝐴𝑁‖
It is proved in the same way that ∆𝐷𝐴𝑀 = ∆𝐵𝐶𝑁 ⟹ ‖𝑀𝐶‖ = ‖𝑁𝐶‖.
Thus 𝐴𝑁𝐶𝑀 is a parallelogram.

33
To determine the angle α:
We write
thus we have established the positions of the lines of the locus.
constant for 𝐴, 𝐵, 𝐶, 𝐷 – fixed points.
We must find the geometrical locus of points 𝑀 such that the ratio of the
distances from this point to two concurrent lines to be constant.

34
‖𝑀𝐸‖
‖𝑀𝐹‖
= 𝑘. Let M' be another point with the same property, namely
‖𝑀′𝐸′‖
‖𝑀′𝐹′‖
= 𝑘.
𝑃, 𝑀, 𝑀′ collinear ⟹ the locus is a line that passes through 𝑃.
When the points are in ∢𝐶𝑃𝐵 we obtain one more line that passes through 𝑃.
Thus the locus is formed by two concurrent lines through 𝑃, from which we
eliminate point 𝑃, because the distances from 𝑃 to both lines are 0 and their ratio is
indefinite.
Vice versa, if points 𝑁 and 𝑁′ are on the same line passing through 𝑃, the ratio
of their distances to lines 𝐴𝐵 and 𝐶𝐷 is constant.
We show in the same way as in the previous problem that:
‖𝑀𝐸‖
‖𝑀𝐹‖
= 𝑘 ⟹
‖𝑀𝐸‖
‖𝑀𝐹‖ + ‖𝑀𝐸‖
=
𝑘
1 − 𝑘
⟹ ‖𝑀𝐸‖ =
𝑘𝑑
1 + 𝑘
,
and the locus of the points which are located at a constant distance from a given
line is a parallel to the respective line, located between the two parallels.
If ||𝐴𝐵|| > ||𝐶𝐷|| ⟹ 𝑑(𝑀𝐴𝐸) < 𝑑(𝑀𝐶𝐷).

35
Then, if
𝑀𝐸
𝑀𝐹
= 𝑘 ⟹
𝑀𝐸
𝑀𝐹 − 𝑀𝐸
=
𝑘
1 − 𝑘
⟹
𝑀𝐸
𝑑
=
𝑘
1 − 𝑘
⟹ 𝑀𝐸 =
𝑘𝑑
1 − 𝑘
,
thus we obtain one more parallel to 𝐴𝐵.
Solution no. 1
We suppose that ABCD is not a parallelogram. Let {𝐼} = 𝐴𝐵 ∩ 𝐶𝐷. We build 𝐸 ∈
(𝐼𝐴 such that 𝐼𝐸 = 𝐴𝐵 and 𝐹 ∈ (𝐼𝐶 such that 𝐼𝐹 = 𝐶𝐷. If 𝑀 a point that verifies
𝜎[𝐴𝐵𝑀] + 𝜎[𝐶𝐷𝑀] = 1 (1), then, because 𝜎[𝐴𝐵𝑀] = 𝜎[𝑀𝐼𝐸] and 𝜎[𝐶𝐷𝑀] = 𝜎[𝑀𝐼𝐹], it
results that 𝜎[𝑀𝐼𝐸] + 𝜎[𝑀𝐼𝐹] = 𝑘 (2).
We obtain that 𝜎[𝑀𝐸𝐼𝐹] = 𝑘.
On the other hand, the points 𝐸, 𝐹 are fixed, therefore 𝜎[𝐼𝐸𝐹] = 𝑘′
= const. That
is, 𝜎[𝑀𝐸𝐹] = 𝑘 − 𝑘′
= const.
Because 𝐸𝐹 = const., we have 𝑑(𝑀, 𝐸𝐹) =
2(𝑘−𝑘′)
𝐸𝐹
= const., which shows that 𝑀
belongs to a line that is parallel to 𝐸𝐹, taken at the distance
2(𝑘−𝑘′)
𝐸𝐹
.
Therefore, the locus points are those on the line parallel to 𝐸𝐹, located inside the
quadrilateral 𝐴𝐵𝐶𝐷. They belong to the segment [𝐸′𝐹′] in Fig. 1.
Reciprocally, if 𝑀 ∈ [𝐸′𝐹′], then 𝜎[𝑀𝐴𝐵] + 𝜎[𝑀𝐶𝐷] = 𝜎[𝑀𝐼𝐸] + 𝜎[𝑀𝐼𝐹] =
𝜎[𝑀𝐸𝐼𝐹] = 𝜎[𝐼𝐸𝐹] + 𝜎[𝑀𝐸𝑃] = 𝑘′
+
𝐸𝐹∙2(𝑘−𝑘′)
2∙𝐸𝐹
= 𝑘.

36
In conclusion, the locus of points 𝑀 inside the quadrilateral 𝐴𝐵𝐶𝐷 which occurs
for relation (1) where 𝑘 is a positive constant smaller than 𝑆 = 𝜎[𝐴𝐵𝐶𝐷] is a line
segment.
If 𝐴𝐵𝐶𝐷 is a trapeze having 𝐴𝐵 and 𝐶𝐷 as bases, then we reconstruct the
reasoning as 𝐴𝐷 ∩ 𝐵𝐶 = {𝐼} and 𝜎[𝑀𝐴𝐷] + 𝜎[𝑀𝐵𝐶] = 𝑠 − 𝑘 = const.
If 𝐴𝐵𝐶𝐷 is a parallelogram, one shows without difficulty that the locus is a
segment parallel to 𝐴𝐵.
Solution no. 2 (Ion Patrascu)
We prove that the locus of points 𝑀 which verify the relationship 𝜎[𝑀𝐴𝐵] +
𝜎[𝑀𝐶𝐷] = 𝑘 (1) from inside the convex quadrilateral 𝐴𝐵𝐶𝐷 of area 𝑠 (𝑘 ⊂ 𝑠) is a line
segment.
Let’s suppose that 𝐴𝐵 ∩ 𝐶𝐷 = {𝐼}, see Fig. 2. There is a point 𝑃 of the locus which
belongs to the line 𝐶𝐷. Therefore, we have (𝑃; 𝐴𝐵) =
2𝑘
𝐴𝐵
. Also, there is the point
𝑄 ∈ 𝐴𝐵 such that 𝑑(𝑄; 𝐶𝐷) =
2𝑘
𝐶𝐷
.
Now, we prove that the points from inside the quadrilateral 𝐴𝐵𝐶𝐷 that are on
the segment [𝑃𝑄] belong to the locus.
Let 𝑀 ∈ int[𝐴𝐵𝐶𝐷] ∩ [𝑃𝑄]. We denote 𝑀1 and 𝑀2 the projections of 𝑀 on 𝐴𝐵 and
𝐶𝐷 respectively. Also, let 𝑃1 be the projection of 𝑃 on 𝐴𝐵 and 𝑄1 the projection of
𝑄 on 𝐶𝐷. The triangles 𝑃𝑄𝑄1 and 𝑃𝑀𝑀2 are alike, which means that
𝑀𝑀2
𝑄𝑄1
=
𝑀𝑃
𝑃𝑄
(2),

37
and the triangles 𝑀𝑀1 𝑄 and 𝑃𝑃1 𝑄 are alike, which means that
𝑀𝑀1
𝑃𝑃1
=
𝑀𝑄
𝑃𝑄
(3).
By adding member by member the relations (2) and (3), we obtain
𝑀𝑀2
𝑄𝑄1
+
𝑀𝑀1
𝑃𝑃1
=
𝑀𝑃 + 𝑀𝑄
𝑃𝑄
= 1 (4).
Substituting in (4), 𝑄𝑄1 =
2𝑘
𝐶𝐷
and 𝑃1 =
2𝑘
𝐴𝐵
, we get 𝐴𝐵 ∙ 𝑀𝑀1 + 𝐶𝐷 ∙ 𝑀𝑀2 = 2𝑘, that
is 𝜎[𝑀𝐴𝐵] + 𝜎[𝑀𝐶𝐷] = 𝑘.
We prove now by reductio ad absurdum that there is no point inside the
quadrilateral 𝐴𝐵𝐶𝐷 that is not situated on the segment [𝑃𝑄], built as shown, to
verify the relation (1).
Let a point 𝑀′
inside the quadrilateral 𝐴𝐵𝐶𝐷 that verifies the relation (1), 𝑀′
∉
[𝑃𝑄]. We build 𝑀′
𝑇 ∩ 𝐴𝐵, 𝑀′
𝑈 ∥ 𝐶𝐷, where 𝑇 and 𝑈 are situated on [𝑃𝑄], see Fig. 3.
We denote 𝑀1
′
, 𝑇1, 𝑈1 the projections of 𝑀1, 𝑇, 𝑈 on 𝐴𝐵 and 𝑀2
′
, 𝑇2, 𝑈2 the
projections of the same points on 𝐶𝐷.
We have the relations:
𝑀′
𝑀1
′
∙ 𝐴𝐵 + 𝑀′
𝑀2
′
∙ 𝐶𝐷 = 2𝑘 (5),
𝑇𝑇1 ∙ 𝐴𝐵 + 𝑇𝑇2 ∙ 𝐶𝐷 = 2𝑘 (6).
Because 𝑀′
𝑀1
′
= 𝑇𝑇1 and 𝑀′
𝑀2
′
= 𝑈𝑈2, substituting in (5), we get:
𝑇𝑇1 ∙ 𝐴𝐵 + 𝑈𝑈2 ∙ 𝐶𝐷 = 2𝑘 (7).
From (6) and (7), we get that 𝑇𝑇2 = 𝑈𝑈2, which drives us to 𝑃𝑄 ∥ 𝐶𝐷, false!

38
Problems in Geometry and Trigonometry
39. Find the locus of the points such that the sum of the distances to two
concurrent lines to be constant and equal to 𝑙.
40. Show that in any triangle 𝐴𝐵𝐶 we have:
a. 𝑏 𝑐𝑜𝑠𝐶 + 𝑐 𝑐𝑜𝑠𝐵 = 𝑎; b. 𝑏 𝑐𝑜𝑠𝐵 + 𝑐 𝑐𝑜𝑠𝐶 = 𝑎 𝑐𝑜𝑠(𝐵 − 𝐶).
41. Show that among the angles of the triangle 𝐴𝐵𝐶 we have:
a. 𝑏 𝑐𝑜𝑠𝐶 − 𝑐 𝑐𝑜𝑠𝐵 =
𝑏2−𝑎2
𝑎
;
b. 2(𝑏𝑐 𝑐𝑜𝑠𝐴 + 𝑎𝑐 𝑐𝑜𝑠𝐵 + 𝑎𝑏 𝑐𝑜𝑠𝐶 = 𝑎2
+ 𝑏2
+ 𝑐2
.
42. Using the law of cosines prove that 4𝑚
2
𝑎
= 2(𝑏2
+ 𝑐2) − 𝑎2
, where 𝑚 𝑎 is the
length of the median corresponding to the side of 𝑎 length.
43. Show that the triangle 𝐴𝐵𝐶 where
𝑎+𝑐
𝑏
= cot
𝐵
2
is right-angled.
44. Show that, if in the triangle 𝐴𝐵𝐶 we have cot 𝐴 + cot 𝐵 = 2 cot 𝐶 ⇒ 𝑎2
+
𝑏2
= 2𝑐2
.
45. Determine the unknown elements of the triangle 𝐴𝐵𝐶, given:
a. 𝐴, 𝐵 and 𝑝;
b. 𝑎 + 𝑏 = 𝑚, 𝐴 and 𝐵;
c. 𝑎, 𝐴; 𝑏 − 𝑐 = 𝑎.

39
46. Show that in any triangle 𝐴𝐵𝐶 we have tan
𝐴−𝐵
2
tan
𝐶
2
=
𝑎−𝑏
𝑎+𝑏
(tangents
theorem).
47. In triangle 𝐴𝐵𝐶 it is given 𝐴̂ = 60° and
𝑏
𝑐
= 2 + √3. Find tan
𝐵−𝐶
2
and angles
𝐵 and 𝐶.
48. In a convex quadrilateral 𝐴𝐵𝐶𝐷, there are given ‖𝐴𝐷‖ = 7(√6 − √2), ‖𝐶𝐷‖ =
13, ‖𝐵𝐶‖ = 15, 𝐶 = arccos
33
65
, and 𝐷 =
𝜋
4
+ arccos
5
13
. The other angles of the
quadrilateral and ‖𝐴𝐵‖ are required.
49. Find the area of ∆𝐴𝐵𝐶 when:
a. 𝑎 = 17, 𝐵 = arcsin
24
25
, 𝐶 = arcsin
12
13
;
b. 𝑏 = 2, 𝐴̂ ∈ 135°, 𝐶̂ ∈ 30°;
c. 𝑎 = 7, 𝑏 = 5, 𝑐 = 6;
d. 𝐴̂ ∈ 18°, 𝑏 = 4, 𝑐 = 6.
50. How many distinct triangles from the point of view of symmetry are there
such that 𝑎 = 15, 𝑐 = 13, 𝑠 = 24?
51. Find the area of ∆𝐴𝐵𝐶 if 𝑎 = √6, 𝐴̂ ∈ 60°, 𝑏 + 𝑐 = 3 + √3.
52. Find the area of the quadrilateral from problem 48.

40
53. If 𝑆 𝑛 is the area of the regular polygon with 𝑛 sides, find:
𝑆3, 𝑆4, 𝑆6, 𝑆8, 𝑆12, 𝑆20 in relation to 𝑅, the radius of the circle inscribed in the
polygon.
54. Find the area of the regular polygon 𝐴𝐵𝐶𝐷 … 𝑀 inscribed in the circle with
radius 𝑅, knowing that:
1
‖𝐴𝐵‖
=
1
‖𝐴𝐶‖
+
1
‖𝐴𝐷‖
.
55. Prove that in any triangle ABC we have:
a. 𝑟 = (𝑝 − 𝑎) tan
𝐴
2
;
b. 𝑆 = (𝑝 − 𝑎) tan
𝐴
2
;
c. 𝑝 = 4𝑅 cos
𝐴
2
cos
𝐵
2
cos
𝐶
2
;
d. 𝑝 − 𝑎 = 4𝑅 cos
𝐴
2
cos
𝐵
2
cos
𝐶
2
;
e. 𝑚 𝑎
2
= 𝑅2
(sin2
𝐴 + 4 cos 𝐴 sin 𝐵 sin 𝐶;
f. ℎ 𝑎 = 2𝑅 sin 𝐵 sin 𝐶.
56. If 𝑙 is the center of the circle inscribed in triangle 𝐴𝐵𝐶 show that ‖𝐴𝐼‖ =
4𝑅 sin
𝐵
2
sin
𝐶
2
.
57. Prove the law of sine using the analytic method.
58. Using the law of sine, show that in a triangle the larger side lies opposite
to the larger angle.
a.
𝑎 cos 𝐶 − 𝑏 cos 𝐵
𝑎 cos 𝐵 − 𝑏 cos 𝐴
+ cos 𝐶 = 0, 𝑎 ≠ 𝑏;

41
b.
sin(𝐴 − 𝐵) sin 𝐶
1 + cos(𝐴 − 𝐵) cos 𝐶
=
𝑎2
− 𝑏2
𝑎2 + 𝑏2
;
c. (𝑎 + 𝑐) cos
𝐵
4
+ 𝑎 cos(𝐴 +
3𝐵
4
) = 2𝑐 cos
𝐵
2
cos
𝐵
4
.
60. In a triangle 𝐴𝐵𝐶, 𝐴 ∈ 45°, ‖𝐴𝐵‖ = 𝑎, ‖𝐴𝐶‖ =
2√2
3
𝑎. Show that tan 𝐵 = 2.
61. Let 𝐴′, 𝐵′, 𝐶′ be tangent points of the circle inscribed in a triangle 𝐴𝐵𝐶 with
its sides. Show that
𝜎[𝐴′ 𝐵′ 𝐶′]
𝜎[𝐴𝐵𝐶]
=
𝑟
2𝑅
.
62. Show that in any triangle 𝐴𝐵𝐶 sin
𝐴
2
≤
𝑎
2√𝑏𝑐
.
63. Solve the triangle 𝐴𝐵𝐶, knowing its elements 𝐴, 𝐵 and area 𝑆.
64. Solve the triangle 𝐴𝐵𝐶, knowing 𝑎 = 13, arc cos
4
5
, and the corresponding
median for side 𝑎, 𝑚 𝑎 =
1
2
√15√3 .
65. Find the angles of the triangle 𝐴𝐵𝐶, knowing that 𝐵 − 𝐶 =
2𝜋
3
and 𝑅 = 8𝑟,
where 𝑅 and 𝑟 are the radii of the circles circumscribed and inscribed in the
triangle.

42
Solutions
Let 𝑑1 and 𝑑2 be the two concurrent lines. We draw 2 parallel lines to 𝑑1 located
on its both sides at distance 𝑙. These intersect on 𝑑2 at 𝐷 and 𝐵, which will be
points of the locus to be found, because the sum of the distances 𝑑(𝐵, 𝑑1) +
𝑑(𝐵, 𝑑2) = 𝑙 + 0 verifies the condition from the statement.
We draw two parallel lines with 𝑑2 located at distance 𝑙 from it, which cut 𝑑1 in 𝐴
and 𝐶, which are as well points of the locus to be found. The equidistant parallel
lines determine on 𝑑2 congruent segments ⟹
|𝐷𝑂| ≡ |𝑂𝐵|
|𝐴𝑂| ≡ |𝑂𝐶|
, in the same way 𝐴𝐵𝐶𝐷
is a parallelogram.
∆𝐵𝑂𝐶,
‖𝐶𝐶′‖ = 𝑑(𝐶, 𝑑2)
‖𝐵𝐵′‖ = 𝑑(𝐵, 𝑑1)
} ⟹ ‖𝐶𝐶′‖ = ‖𝐵𝐵′‖
⟹ ∆𝐵𝑂𝐶 is isosceles.
⟹ ||𝑂𝐶|| = ||𝑂𝐵|| ⟹ 𝐴𝐵𝐶𝐷 is a rectangle. Any point 𝑀 we take on the sides of this
rectangle, we have ||𝑅1, 𝑑1|| + ||𝑀, 𝑑2|| = 𝑙, using the propriety according to which
the sum of the distances from a point on the base of an isosceles triangle at the
sides is constant and equal to the height that starts from one vertex of the base,
namely 𝑙. Thus the desired locus is rectangle 𝐴𝐵𝐶𝐷.

43
𝑚 𝑎
2
= 𝑐2
+
𝑎2
4
− 2
𝑎
2
𝑐 cos 𝐵 ;
4𝑚 𝑎
2
= 4𝑐2
+ 𝑎2
− 4𝑎𝑐 cos 𝐵 = 4𝑐2
+ 𝑎2
− 4𝑎𝑐
𝑎2
+ 𝑐2
− 𝑏2
2𝑎𝑐
= 4𝑐2
+ 𝑎2
− 2𝑎 − 2𝑐2
+ 2𝑏2
= 2𝑐2
+ 2𝑏2
− 𝑎2
= 2(𝑏2
+ 𝑐2) − 𝑎2
.

44
Using the sine theorem, 𝑎 = 𝑚 sin 𝐴.
𝐴 − 𝐶
2
= −
𝐵
2
⟹ 𝐴 − 𝐵 = 𝐶 or 𝐴 − 𝐶 = 𝐵 ⟹
𝐴 = 𝐵 + 𝐶 2𝐴 = 1800
𝐴 = 900
or ⟹ or ⟹ or
𝐴 + 𝐵 = 𝐶 2𝐶 = 1800
𝐶 = 900
By substitution:

45
a. Using the law of sine,
b.
c.
Therefore,
We solve the system, and find 𝐵 and 𝐶. Then we find 𝑏 =
𝑎 sin 𝐵
sin 𝐴
and 𝑐 = 𝑏 − 𝑑.
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
= 𝑚 ⟹
𝑎 = 𝑚 sin 𝐴
𝑏 = 𝑚 sin 𝐵
;
𝑎 − 𝑏
𝑎 + 𝑏
=
𝑚 sin 𝐴 − 𝑚 sin 𝐵
𝑚 sin 𝐴 + 𝑚 sin 𝐵
=
sin 𝐴 − sin 𝐵
sin 𝐴 + sin 𝐵
=
2 sin
𝐴 − 𝐵
2
cos
𝐴 + 𝐵
2
2 sin
𝐴 + 𝐵
2 cos
𝐴 − 𝐵
2
= tan
𝐴 − 𝐵
2
sin
𝐶
2
cos
𝐶
2
= tan
𝐴 − 𝐵
2
tan
𝐶
2
.

46
Using tangents’ theorem,
So
In ∆𝐵𝐷𝐶 we have

47
In ∆𝐴𝐷𝐵,
In ∆𝐴𝐷𝐵 we apply sine’s theorem:
Or we find 𝜇(𝐷𝐵𝐶̂) and we add it to
𝜋
6
.

48
sin 𝐴 =
−1+√5
4
, because 𝑚(𝐴) < 1800
and sin 𝐴 > 0.

49

50
At problem 9 we’ve found that
With Heron’s formula, we find the area of each triangle and we add them up.
The formula for the area of a regular polygon:

51
In ∆𝐵𝑂𝑀:
In ∆𝑁𝑂𝐶:
In ∆𝑃𝑂𝐷:
Substituting (1), (2), (3) in the given relation:
or
which is impossible.
𝑛 =
𝑚(complete circle)
𝑚(𝐴𝐵̂ )
=
2𝜋
2𝜋
7
= 7.
Thus the polygon has 7 sides.

52

53
We apply the law of sine in ∆𝐴𝐵𝐼:
The law of sine applied in ∆𝐴𝐵𝐶:
In ∆𝐴𝐶𝐶′: sin(1800
− 𝐴) =
‖𝐶𝐶′‖
𝑏
⟹ ‖𝐶𝐶′‖ = 𝑏 sin 𝐴 ; cos(1800
− 𝐴) = 𝑏 cos 𝐴.
So the coordinates of 𝐶 are (−𝑏 cos 𝐴 , 𝑏 sin 𝐴).
The center of the inscribed circle is at the intersection of the perpendicular lines
drawn through the midpoints of sides 𝐴𝐵 and 𝐴𝐶.
The equation of the line 𝐸𝑂:

54
If we redo the calculus for the same draw, we have the following result:
(𝑏 cos 𝐴 , 𝑏 sin 𝐴).
using the law of cosine.
𝑎
sin 𝐴
=
𝑏
sin 𝐵
=
𝑐
sin 𝐶
= 2𝑅.
We suppose that 𝑎 > 𝑏. Let’s prove that 𝐴 > 𝐵.
𝑎
sin 𝐴
=
𝑏
sin 𝐵
⇒
𝑎
𝑏
=
sin 𝐴
sin 𝐵
𝑎 > 𝑏 ⇒
𝑎
𝑏
> 1
} ⟹
sin 𝐴
sin 𝐵
> 1 ⟹ 𝐴, 𝐵, 𝐶 ∈ (0, 𝜋) ⟹ sin 𝐵 > 0 ⟹ sin 𝐴 > sin 𝐵
⟹ sin 𝐴 − sin 𝐵 > 0 ⟹ 2 sin
𝐴 − 𝐵
2
cos
𝐴 + 𝐵
2
> 0 ⟹
𝐴 + 𝐵
2
=
1800
− 𝐶
2
= 900
−
𝐶
2
.

55
cos
𝐴+𝐵
2
= cos (90circ
−
𝐶
2
) = sin
𝐶
2
> 0, therefore
𝐴−𝐵
2
> 0 ⟹ 𝐴 > 𝐵;
(−
𝜋
2
<
𝐴 − 𝐵
2
<
𝜋
2
).
b. We transform the product into a sum:
From (1) and (2) ⟹
We consider the last two terms:

56
We apply the law of cosines in triangle ABC:
‖𝐼𝐴‖ = ‖𝐼𝐵‖ = ‖𝐼𝐶‖ = 𝑟
𝐼𝐶′ ⊥ 𝐴𝐵
𝐼𝐴′ ⊥ 𝐵𝐶
} ⇒ 𝐼𝐴′𝐵′𝐶′ inscribable quadrilateral
𝑚(𝐴′𝐼𝐶′) = 180 − 𝐵̂ ⟹ sin(𝐴′𝐼𝐶′̂ ) = sin 𝐵̂
Similarly, 𝐴′𝐼𝐵′̂ = sin 𝐶 and 𝐶′𝐼𝐵′̂ = sin 𝐴.

57
In the same way,
0 < 𝐴 ⇒<
𝐴
2
<
𝜋
2
⇒ sin
𝐴
2
> 0 ;
sin
𝐴
2
= √
1 − cos 𝐴
2
⟹
sin2
𝐴
2
=
1 − cos 𝐴
2
cos 𝐴 =
𝑏2
+ 𝑐2
− 𝑎2
2𝑏𝑐
}
⟹ sin2
𝐴
2
=
1 −
𝑏2
− 𝑎2
+ 𝑐2
2𝑏𝑐
2
=
𝑎2
− (𝑏 − 𝑐)2
4𝑏𝑐
≤
𝑎2
4𝑏𝑐
⟹ sin
𝐴
2
≤
𝑎
2√𝑏𝑐
.
𝐶 = 𝜋 − (𝐴 + 𝐵)
𝑆 =
𝑎2 sin 𝐵 sin 𝐶
2 sin 𝐴
𝐴, 𝐵, 𝐶 are known
} ⟹ We find 𝑎.
𝑎
sin 𝑎
=
𝑏
sin 𝑏
⟹ 𝑏 =
𝑎 sin 𝐵
sin 𝐴
. In the same way, we find 𝑐.

58
𝑏2
+ 𝑐2
= 841
𝑏𝑐 = 420
} ⟹ {
𝑏 = 21
𝑐 = 20
or {
𝑏 = 20
𝑐 = 21
We find 𝐵.
We find the sum.
Or
We already know that
𝑟
𝑅
= 4 sin
𝐴
2
sin
𝐵
2
sin
𝐶
2
We write sin
𝐴
2
= 𝑡. We have

59
From this system we find 𝐵 and 𝐶.

60
Other Problems in Geometry
and Trigonometry (10th
grade)
66. Show that a convex polygon can’t have more than three acute angles.
67. Let 𝐴𝐵𝐶 be a triangle. Find the locus of points 𝑀 ∈ (𝐴𝐵𝐶), for which
𝜎[𝐴𝐵𝑀] = 𝜎[𝐴𝐶𝑀].
68. A convex quadrilateral 𝐴𝐵𝐶𝐷 is given. Find the locus of points 𝑀 ∈
𝑖𝑛𝑡. 𝐴𝐵𝐶𝐷, for which 𝜎[𝑀𝐵𝐶𝐷] = 𝜎[𝑀𝐵𝐴𝐷].
69. Determine a line 𝑀𝑁, parallel to the bases of a trapezoid 𝐴𝐵𝐶𝐷 (𝑀 ∈
|𝐴𝐷|, 𝑁 ∈ |𝐵𝐶|) such that the difference of the areas of [𝐴𝐵𝑁𝑀] and [𝑀𝑁𝐶𝐷]
to be equal to a given number.
70. On the sides of ∆𝐴𝐵𝐶 we take the points 𝐷, 𝐸, 𝐹 such that
𝐵𝐷
𝐷𝐶
=
𝐶𝐸
𝐸𝐴
=
𝐴𝐹
𝐹𝐵
= 2.
Find the ratio of the areas of triangles 𝐷𝐸𝐹 and 𝐴𝐵𝐶.
71. Consider the equilateral triangle 𝐴𝐵𝐶 and the disk [𝐶 (𝑂,
𝑎
3
)], where 𝑂 is the
orthocenter of the triangle and 𝑎 = ‖𝐴𝐵‖. Determine the area [𝐴𝐵𝐶] −
[𝐶 (𝑂,
𝑎
3
)].

61
a. 1 + cos 𝐴 cos(𝐵 − 𝐶) =
𝑏2+𝑐2
4𝑅2 ;
b. (𝑏2
+ 𝑐2
= 𝑎2) tan 𝐴 = 4𝑆;
c.
𝑏+𝑐
2𝑐 cos
𝐴
2
=
sin(
𝐴
2
+𝐶)
sin(𝐴+𝐵)
;
d. 𝑝 = 𝑟 (cot
𝐴
2
+ cot
𝐵
2
+ cot 𝐶2);
e. cot
𝐴
2
+ cot
𝐵
2
+ cot
𝐶
2
=
𝑝
𝑟
.
73. If 𝐻 is the orthocenter of triangle 𝐴𝐵𝐶, show that:
a. ‖𝐴𝐻‖ = 2𝑅 cos 𝐴;
b. 𝑎‖𝐴𝐻‖ + 𝑏‖𝐵𝐻‖ + 𝑐‖𝐶𝐻‖ = 4𝑆.
74. If 𝑂 is the orthocenter of the circumscribed circle of triangle 𝐴𝐵𝐶 and 𝐼 is
the center of the inscribed circle, show that ‖𝑂𝐼‖2
= 𝑅(𝑅 – 2𝑟).
75. Show that in any triangle 𝐴𝐵𝐶 we have: cos2 𝐵−𝐶
2
≥
2𝑟
𝑅
.
76. Find 𝑧 𝑛
+
1
𝑧 𝑛
knowing that 𝑧 +
1
𝑧
= 2 sin 𝛼.
77. Solve the equation: (𝑧 + 1) 𝑛
− (𝑧 − 1) 𝑛
= 0.
78. Prove that if 𝑧 <
1
2
then |(1 + 𝑖)𝑧3
+ 𝑖𝑧| ≤
3
4
.

62
79. One gives the lines 𝑑 and 𝑑′. Show that through each point in the space
passes a perpendicular line to 𝑑 and 𝑑′.
80. There are given the lines 𝑑 and 𝑑′, which are not in the same plane, and
the points 𝐴 ∈ 𝑑, 𝐵 ∈ 𝑑′. Find the locus of points 𝑀 for which pr 𝑑 𝑀 = 𝐴 and
pr 𝑑′ 𝑀 = 𝐵.
81. Find the locus of the points inside a trihedral angle 𝑎𝑏𝑐̂ equally distant
from the edges of 𝑎, 𝑏, 𝑐.
82. Construct a line which intersects two given lines and which is perpendicular
to another given line.
83. One gives the points 𝐴 and 𝐵 located on the same side of a plane; find in
this plane the point for which the sum of its distances to 𝐴 and 𝐵 is
minimal.
84. Through a line draw a plane onto which the projections of two lines to be
parallel.
85. Consider a tetrahedron [𝐴𝐵𝐶𝐷] and centroids 𝐿, 𝑀, 𝑁 of triangles
𝐵𝐶𝐷, 𝐶𝐴𝐷, 𝐴𝐵𝐷.
a. Show that (𝐴𝐵𝐶) ∥ (𝐿𝑀𝑁);
b. Find the ratio
𝜎[𝐴𝐵𝐶]
𝜎[𝐿𝑀𝑁]
.

63
86. Consider a cube [𝐴𝐵𝐶𝐷𝐴′𝐵′𝐶′𝐷′]. The point 𝐴 is projected onto 𝐴′𝐵, 𝐴′𝐶, 𝐴′𝐷
respectively in 𝐴1, 𝐴2, 𝐴3. Show that:
a. 𝐴′
𝐶 ⊥ (𝐴1 𝐴2 𝐴3);
b. 𝐴𝐴1 ⊥ 𝐴1 𝐴2, 𝐴𝐴3 ⊥ 𝐴3 𝐴2;
c. 𝐴𝐴1 𝐴2 𝐴3 is an inscribable quadrilateral.
87. Consider the right triangles 𝐵𝐴𝐶 and 𝐴𝐵𝐷 (𝑚(𝐵𝐴𝐶̂)) = 𝑚((𝐴𝐵𝐷)̂ = 900
)
located on perpendicular planes 𝑀 and 𝑁, being midpoints of segments
[𝐴𝐵], [𝐶𝐷]. Show that 𝑀𝑁 ⊥ 𝐶𝐷.
88. Prove that the bisector half-plane of a dihedral angle inside a tetrahedron
divides the opposite edge in proportional segments with the areas of the
adjacent faces.
89. Let 𝐴 be a vertex of a regular tetrahedron and 𝑃, 𝑄 two points on its
surface. Show that 𝑚(𝑃𝐴𝑄̂) ≤ 600
.
90. Show that the sum of the measures of the dihedral angles of a tetrahedron
is bigger than 360°.
91. Consider lines 𝑑1, 𝑑2 contained in a plane 𝛼 and a line 𝐴𝐵 which intersects
plane 𝛼 at point 𝐶. A variable line, included in 𝛼 and passing through 𝐶 all
𝑑1, 𝑑2 respectively at 𝑀𝑁. Find the locus of the intersection 𝐴𝑀 ∩ 𝐵𝑁. In
which case is the locus an empty set?

64
92. A plane 𝛼 intersects sides [𝐴𝐵], [𝐵𝐶], [𝐶𝐷], [𝐷𝐴] of a tetrahedron [𝐴𝐵𝐶𝐷] at
points 𝐿, 𝑀, 𝑁, 𝑃. Prove that ‖𝐴𝐿‖ ∙ ‖𝐵𝑀‖ ∙ ‖𝐶𝑁‖ ∙ ‖𝑃𝐷‖ = ‖𝐵𝐿‖ ∙ ‖𝐶𝑀‖ ∙
‖𝐷𝑁‖ ∙ ‖𝐴𝑃‖.
93. From a point 𝐴 located outside a plane 𝛼, we draw the perpendicular line
𝐴𝑂, 𝑂 ∈ 𝛼, and we take 𝐵, 𝐶 ∈ 𝛼. Let 𝐻, 𝐻1 be the orthocenters of triangles
𝐴𝐵𝐶, 𝑂𝐵𝐶; 𝐴𝐷 and 𝐵𝐸 heights in triangle 𝐴𝐵𝐶; and 𝐵𝐸1 height in triangle
𝑂𝐵𝐶. Show that:
a. 𝐻𝐻1 ⊥ (𝐴𝐵𝐶);
b. ‖
𝑂𝐴
𝐴𝐷
‖ ∙ ‖
𝐷𝐻1
𝐻1 𝐵
‖ ∙ ‖
𝐵𝐸
𝐸𝐸1
‖ = 1.
94. Being given a tetrahedron [𝐴𝐵𝐶𝐷] where 𝐴𝐵 ⊥ 𝐶𝐷 and 𝐴𝐶 ⊥ 𝐵𝐷, show that:
a. ‖𝐴𝐵‖2
+ ‖𝐶𝐷‖2
= ‖𝐵𝐶‖2
+ ‖𝐴𝐷‖2
= ‖𝐶𝐴‖2
+ ‖𝐵𝐷‖2
;
b. The midpoints of the 6 edges are located on a sphere.
95. It is given a triangular prism [𝐴𝐵𝐶𝐴′𝐵′𝐶′] which has square lateral faces. Let
𝑀 be a mobile point [𝐴𝐵′], 𝑁 the projection of 𝑀 onto (𝐵𝐶𝐶′) and 𝐴ʺ the
midpoint of [𝐵′𝐶ʺ]. Show that 𝐴′𝑁 and 𝑀𝐴ʺ intersect in a point 𝑃 and find
the locus of 𝑃.
96. We have the tetrahedron [𝐴𝐵𝐶𝐷] and let 𝐺 be the centroid of triangle
𝐵𝐶𝐷. Show that if 𝑀 ∈ 𝐴𝐺 then 𝜐[𝑀𝐺𝐵𝐶] = 𝜐[𝑀𝐺𝐶𝐷] = 𝜐[𝑀𝐺𝐷𝐵].
97. Consider point 𝑀 ∈ the interior of a trirectangular tetrahedron with its
vertex in 𝑂. Draw through 𝑀 a plane which intersects the edges of the

65
respective tetrahedron in points 𝐴, 𝐵, 𝐶 so that 𝑀 is the orthocenter of
∆𝐴𝐵𝐶.
98. A pile of sand has as bases two rectangles located in parallel planes and
trapezoid side faces. Find the volume of the pile, knowing the dimensions
𝑎′, 𝑏′ of the small base, 𝑎, 𝑏 of the larger base, and ℎ the distance between
the two bases.
99. A pyramid frustum is given, with its height ℎ and the areas of the bases 𝐵
and 𝑏. Unite any point 𝜎 of the larger base with the vertices 𝐴, 𝐵, 𝐴′, 𝐵′ of a
side face. Show that 𝜐[𝑂𝐴′𝐵′𝐴] =
√6
√𝐵
𝜐[𝑂𝐴𝐵𝐵′].
100. A triangular prism is circumscribed to a circle of radius 𝑅. Find the area
and the volume of the prism.
101. A right triangle, with its legs 𝑏 and 𝑐 and the hypotenuse 𝑎, revolves by
turns around the hypotenuse and the two legs, 𝑉1, 𝑉2, 𝑉3; 𝑆1, 𝑆2, 𝑆3 being the
volumes, respectively the lateral areas of the three formed shapes, show
that:
a.
1
𝑉1
2 =
1
𝑉2
2 =
1
𝑉3
2;
b.
𝑆2
𝑆3
+
𝑆3
𝑆2
=
𝑆2+𝑆3
𝑆1
.
102. A factory chimney has the shape of a cone frustum and 10m height, the
bases of the cone frustum have external lengths of 3,14m and 1,57m, and
the wall is 18cm thick. Calculate the volume of the chimney.

66
103. A regular pyramid, with its base a square and the angle from the peak of
a side face of measure 𝛼 is inscribed in a sphere of radius 𝑅. Find:
a. the volume of the inscribed pyramid;
b. the lateral and total area of the pyramid;
c. the value 𝛼 when the height of the pyramid is equal to the radius of
the sphere.

67
Solutions
Let 𝐴1, 𝐴2 … 𝐴 𝑛 the vertices of the convex polygon. Let’s assume that it has four
acute angles. The vertices of these angles form a convex quadrilateral 𝐴𝑙 𝐴 𝑘 𝐴 𝑚 𝐴 𝑛.
Due to the fact that the polygon is convex, the segments |𝐴𝑙 𝐴 𝑘|, |𝐴 𝑘 𝐴 𝑚|, |𝐴 𝑚 𝐴 𝑛|,
|𝐴 𝑛 𝐴𝑙| are inside the initial polygon. We find that the angles of the quadrilateral are
acute, which is absurd, because their sum is 360°.
Another solution: We assume that 𝐴𝑙 𝐴 𝑘 𝐴 𝑚 𝐴 𝑛 is a convex polygon with all its
angles acute ⟹ the sum of the external angles is bigger than 360°, which is absurd
(the sum of the measures of the external angles of a convex polygon is 360°).

68
Let |𝐴𝐴′| be the median from 𝐴 and 𝐶𝑄 ⊥ 𝐴𝐴′
, 𝐵𝑃 ⊥ 𝐴𝐴′.
∆𝐵𝐴′𝑃 ≡ 𝐶𝐴′𝑄 because:
{
𝑃𝐵𝐶̂ ≡ 𝐵𝐶𝑄̂ alternate interior
𝑃𝐴′𝐵̂ ≡ 𝐶𝐴′ 𝑄̂ vertical angles
𝐵𝐴′
≡ 𝐴′
𝐶
⟹ ||𝐵𝑃|| = ||𝑄𝐶|| and by its construction 𝐵𝑃 ⊥ 𝐴𝐴′, 𝐶𝑄 ⊥ 𝐴𝐴′.
The desired locus is median |𝐴𝐴′|. Indeed, for any 𝑀 ∈ |𝐴𝐴′| we have 𝜎[𝐴𝐵𝑀] =
𝜎[𝐴𝐶𝑀], because triangles 𝐴𝐵𝑀 and 𝐴𝐶𝑀 have a common side |𝐴𝑀| and its
corresponding height equal ||𝐵𝑃|| = ||𝑄𝐶||.
Vice-versa. If 𝜎[𝐴𝐵𝑀] = 𝜎[𝐴𝐶′𝑀], let’s prove that 𝑀 ∈ |𝐴𝐴′|.
Indeed: 𝜎[𝐴𝐵𝑀] 𝜎[𝐴𝐶𝑀] ⇒ 𝑑 (𝐵, 𝐴𝑀) = 𝑑(𝐶, 𝐴𝑀), because |𝐴𝑀| is a common
side, 𝑑 (𝐵, 𝐴𝑀) = ||𝐵𝑃|| and 𝑑(𝐶, 𝐴𝑀) = ||𝐶𝑄|| and both are perpendicular to 𝐴𝑀 ⟹
𝑃𝐵𝑄𝐶 is a parallelogram, the points 𝑃, 𝑀, 𝑄 are collinear (𝑃, 𝑄 the feet of the
perpendicular lines from 𝐵 and 𝐶 to 𝐴𝑀).
In parallelogram 𝑃𝐵𝑄𝐶 we have |𝑃𝑄| and |𝐵𝐶| diagonals ⟹ 𝐴𝑀 passes through
the middle of |𝐵𝐶|, so 𝑀 ∈ |𝐴𝐴′|, the median from 𝐴.
Let 𝑂 be the midpoint of diagonal |𝐴𝐶| ⟹ ‖𝐴𝑂‖ = ‖𝑂𝐶‖.
𝜎[𝐴𝑂𝐷] = 𝜎[𝐶𝑂𝐷] (1)
Because {
‖𝐴𝑂‖ = ‖𝑂𝐶‖
‖𝑂𝐷′‖ common height
𝜎[𝐴𝑂𝐵] = 𝜎[𝐶𝑂𝐵] (2)

69
the same reasons; we add up (1) and (2) ⟹
𝜎[𝐴𝐷𝑂𝐵] = 𝜎[𝐷𝐶𝐵𝑂] (3),
so 𝑂 is a point of the desired locus.
We construct through 𝑂 a parallel to 𝐵𝐷 until it cuts sides |𝐵𝐶| and |𝐷𝐶| at 𝑃
respectively 𝑄. The desired locus is |𝑃𝑄|.
Indeed (∀)𝑀 ∈ |𝑃𝑄| we have:
𝜎[𝐵𝐷𝑂] = 𝜎[𝐵𝐷𝑀] because 𝑀 and 𝑄 belongs to a parallel to 𝐵𝐷.
𝐵, 𝐷 ∈ a parallel to 𝑂𝑀.
So
𝜎[𝑀𝐵𝐴𝐷] = 𝜎[𝐴𝐵𝑂𝐷]
and 𝜎[𝐵𝐶𝐷𝑀] = 𝜎[𝑂𝐵𝐶𝐷]
and from (3)
} ⟹ 𝜎[𝑀𝐵𝐴𝐷] = 𝜎[𝐵𝐶𝐷𝑀].
Vice-versa: If 𝜎[𝑀𝐵𝐶𝐷] = 𝜎[𝑀𝐵𝐴𝐷], let’s prove that 𝑀 ∈ parallel line through 𝑂 to
𝐵𝐷. Indeed:
𝜎[𝐵𝐶𝐷𝑀] = 𝜎[𝑀𝐵𝐴𝐷]
and because 𝜎[𝐵𝐶𝐷𝑀] + 𝜎[𝑀𝐵𝐴𝐷] = 𝜎[𝐴𝐵𝐶𝐷]
} ⟹ 𝜎[𝑀𝐵𝐶𝐷] = 𝜎[𝑀𝐵𝐴𝐷] =
𝜎[𝐴𝐵𝐶𝐷]
2
(2).
So, from (1) and (2) ⟹ 𝜎[𝑀𝐵𝐴𝐷] = 𝜎[𝐴𝐵𝑂𝐷] ⟹ 𝜎[𝐴𝐵𝐷] + 𝜎[𝐵𝐷𝑂] = 𝜎[𝐴𝐵𝐷] +
𝜎[𝐵𝐷𝑀] ⟹ 𝜎[𝐵𝐷𝑀] ⟹ 𝑀 and 𝑂 are on a parallel to 𝐵𝐷.
We write ||𝐸𝐴|| = 𝑎 and ||𝐸𝐷|| = 𝑏, ||𝐸𝑀|| = 𝑥.

70
We subtract (2) from (3)
We subtract (2) from (4)
From the relation (3), by writing [𝐴𝐵𝐶𝐷] − 𝑆 ⟹ 𝜎[𝐸𝐶𝐷] =
𝑆𝑏2
𝑎2−𝑏2 .
We substitute this in the relation of 𝑥² and we obtain:
and taking into consideration that ||𝐸𝑀|| = ||𝐷𝑀|| + 𝑏, we have
so we have the position of point 𝑀 on the segment |𝐷𝐴| (but it was sufficient to
find the distance ||𝐸𝑀||).
We remark from its construction that 𝐸𝑄||𝐴𝐵||𝑅𝐷, more than that, they are
equidistant parallel lines. Similarly, 𝐸𝑄, 𝑃𝐷, 𝐴𝐶 and 𝐴𝐵, 𝐸𝑄, 𝑅𝐷 are also equidistant
parallel lines.
from the hypothesis

71
We write 𝜎[𝐵𝐹𝑄] = 𝑆.
Based on the following properties:
 two triangles have equal areas if they have equal bases and the same
height;
 two triangles have equal areas if they have the same base and the third
peak on a parallel line to the base,
we have:
by addition
⇒
So
𝜎[𝐷𝐸𝐹]
𝜎[𝐴𝐵𝐶]
=
3𝑆
9𝑆
=
1
3
.
(If necessary the areas 𝑆 can be arranged).
‖𝑂𝐵‖ =
𝑎√3
6
(𝐵𝐵′ median)
In ∆𝑀𝑂𝐵′:
So (𝑀𝑂𝑁̂ ) =
𝜋
3
.
We mark with 𝛴 the disk surface bordered by a side of the triangle outside the
triangle.

72
𝜎[𝛴] = 𝜎[circle sector 𝑀𝑂𝑁] − 𝜎[𝑀𝑂𝑁]
=
𝜋𝑎2
9 ∙ 6
−
𝑎2
9 ∙ 2
sin 600
=
𝜋𝑎2
9 ∙ 6
−
𝑎2
√3
4 ∙ 9
=
𝑎2
18
∙ (
𝜋
3
−
√3
2
).
If through the disk area we subtract three times 𝜎[𝛴], we will find the area of the
disk fraction from the interior of 𝐴𝐵𝐶. So the area of the disk surface inside 𝐴𝐵𝐶 is:
The desired area is obtained by subtracting the calculated area form 𝜎[𝐴𝐵𝐶].
So:
a. 1 + cos 𝐴 ∙ cos(𝐵 − 𝐶) =
𝑏2+𝑐2
4𝑅2
b. We prove that tan 𝐴 =
4𝑆
𝑏2+𝑐2−𝑎2
.

73
⟹ All terms reduce.
d. Let’s prove that cot
𝐴
2
+ cot
𝐵
2
+ cot 𝐶2 =
𝑝
𝑟
.
Indeed
We now have to prove that:

74
a. In triangle 𝐴𝐵𝐵′: ‖𝐴𝐵′‖ = 𝑐 cos 𝐴
In triangle 𝐴𝐻𝐵′:
We used:
Using the power of point 𝐼 in relation to circle 𝐶(𝑂, 𝑅)
Taking into consideration (1), we have ‖𝐼𝐴‖ ∙ ‖𝐼𝐷‖ = 𝑅2
− ‖𝑂𝐼‖².
We now find the distances ||IA|| and ||ID||
In triangle ∆𝐼𝐴𝑃,
We also find ‖𝐼𝐷‖: 𝜇(𝐵𝐼𝐷̂ ) = 𝜇(𝐷𝐵𝐼̂ ) have the same measure, more exactly:

75
In ∆𝐴𝐵𝐷 according to the law of sine, we have:
So taking into consideration (3),
Returning to the relation ‖𝐼𝐴‖ ∙ ‖𝐼𝐷‖ = 𝑅2
− ‖𝑂𝐼‖². with (2) and (4) we have:
Note. We will have to show that
Indeed:
(by Heron’s formula).

76
So:
We calculate for 𝑧1 and 𝑧2:
so 𝑧 𝑛
+
1
𝑧 𝑛
takes the same value for 𝑧1 and for 𝑧2 and it is enough if we
calculate it for 𝑧1.
Analogously:

77
(we substitute −1 with 𝑖² at denominator)
We construct 𝛼 ⊥ 𝑑 and 𝐴 ∈ 𝛼. The so constructed plane is unique. Similarly we
construct 𝛽 ⊥ 𝑑′ and 𝐴 ⊥ 𝛽, 𝛼 ∩ 𝛽 = 𝑎 ∋ 𝐴.
From
𝛼 ⊥ 𝑑 ⇒ 𝑑 ⊥ 𝑎
𝛽 ⊥ 𝑑′ ⇒ 𝑑′ ⊥ 𝑎
} ⟹ 𝑎 is a line which passes through 𝐴 and is perpendicular
to 𝑑 and 𝑑′. The line 𝑎 is unique, because 𝛼 and 𝛽 constructed as above are unique.
We construct plane 𝛼 such that 𝐴 ∈ 𝛼 and 𝑑 ⊥ 𝛼. We construct plane 𝛽 such that
𝐵 ∈ 𝛽 and 𝑑′ ⊥ 𝛽.

78
The so constructed planes 𝛼 and 𝛽 are unique.
Let 𝑎 = 𝛼 ∩ 𝛽 ⟹ 𝑎 ⊂ 𝛼 so (∀) 𝑀 ∈ 𝑎 has the property pr 𝑑 𝑀 = 𝐴.
𝛼 ⊂ 𝛽 ⟹ (∀) 𝑀 ∈ 𝑎 has the property 𝑝𝑟 𝑑′ 𝑀 = 𝐵.
Vice-versa. If there is a point 𝑀 in space such that pr 𝑑 𝑀 = 𝐴 and 𝑝𝑟 𝑑′ 𝑀 = 𝐵 ⟹
𝑀 ∈ 𝑎 and 𝑀 ∈ 𝛽 ⟹ 𝑀 ∈ 𝛼 ∩ 𝛽 ⟹ 𝑀 ∈ 𝑎 (𝛼 and 𝛽 previously constructed).
Let 𝐴 ∈ 𝑎, 𝐵 ∈ 𝑏, 𝐶 ∈ 𝑐 such that ‖𝑂𝐴‖ = ‖𝑂𝐵‖ = ‖𝑂𝐶‖. Triangles 𝑂𝐴𝐵, 𝑂𝐵𝐶, 𝑂𝐴𝐶
are isosceles. The mediator planes of segments ‖𝐴𝐵‖, ‖𝐴𝐶‖, ‖𝐵𝐶 ‖ pass through 𝑂
and 𝑂′ (the center of the circumscribed circle of triangle 𝐴𝐵𝐶). Ray |𝑂𝑂′| is the
desired locus.
Indeed (∀) 𝑀 ∈ |𝑂𝑂′| ⟹ 𝑀 ∈ mediator plane of segments |𝐴𝐵|, |𝐴𝐶| and |𝐵𝐶| ⟹
𝑀 is equally distant from 𝑎, 𝑏 and 𝑐.
Vice-versa: (∀) 𝑀 with the property: 𝑑(𝑀, 𝑎) = 𝑑(𝑀, 𝑏) = 𝑑(𝑀, 𝑐) ⟹ 𝑀 ∈ mediator
plan, mediator planes of segments |𝐴𝐵|, |𝐴𝐶| and |𝐵𝐶| ⟹ 𝑀 ∈ the intersection of
these planes ⟹ 𝑀 ∈ |𝑂𝑂′|.

79
Let 𝑎, 𝑏, 𝑐 be the 3 lines in space.
I. We assume 𝑎 ⊥ 𝑐 and 𝑏 ⊥ 𝑐. Let 𝛼 be a plane such that:
The construction is possible because ⊥ 𝑐 and 𝑏 ⊥ 𝑐. Line 𝐴𝐵 meets 𝑎 on 𝑝 and it is
perpendicular to 𝑐, because 𝐴𝐵 ⊂ 𝛼 and 𝑐 ⊥ 𝛼.
II. If 𝑎 ⊥ 𝑐 or 𝑏 ⊥ 𝑐, the construction is not always possible, only if plane 𝑝(𝑎, 𝑏)
is perpendicular to 𝑐.
III. If 𝑎 ⊥ 𝑐 and 𝑏 ⊥ 𝑐, we construct plane 𝑎 ⊥ 𝑐 so that 𝑎 ⊂ 𝛼 and 𝑏 ⊂ 𝛼 ≠ ∅. Any
point on line a connected with point 𝑏 ∩ 𝛼 is a desired line.
We construct 𝐴′ the symmetrical point of 𝐴 in relation to 𝛼. 𝐴′ and 𝐵 are on
different half-spaces, 𝛼 ∩ |𝐴′
𝐵| = 𝑂.

80
𝑂 is the desired point, because ||𝑂𝐴|| + ||𝑂𝐵|| = ||𝑂𝐴′|| + ||𝑂𝐵|| is minimal when
𝑂 ∈ |𝐴′𝐵|, thus the desired point is 𝑂 = |𝐴′𝐵| ∩ 𝛼.
Let 𝑎, 𝑏, 𝑑 be the 3 given lines and through 𝑑 we construct a plane in which 𝑎 and
𝑏 to be projected after parallel lines.
Let 𝐴 be an arbitrary point on 𝑎. Through 𝐴 we construct line 𝑏′||𝑏. It results from
the figure 𝑏||𝛼, 𝛼 = 𝑝(𝑎, 𝑏′).
Let 𝛽 such that 𝑑 ⊂ 𝛽 and 𝛽 ⊥ 𝛼.
Lines 𝑎 and 𝑏′ are projected onto 𝛽 after the same line 𝑐. Line 𝑏 is projected onto
𝛽 after 𝑏1 and 𝑏1 ∥ 𝑐.
If 𝑏1 ∦ 𝑐,
absurd because 𝑏||𝛼(𝑏||𝑏′).
𝑀 is the centroid in ∆𝐴𝐶𝐷 ⟹
⟹
|𝑀𝐷|
|𝑀𝑃|
= 2 (1)

81
𝑁 is the centroid in ∆𝐴𝐵𝐷 ⟹
⟹
|𝑁𝐷|
|𝑁𝑄|
= 2 (2)
𝐿 is the centroid in ∆𝐵𝐶𝐷 ⟹
⟹
|𝐿𝐷|
|𝐿𝑆|
= 2 (3)
From 1 and 2,
and from 2 and 3
because:
So
𝐵𝐷 ⊥ (𝐴𝐴′𝐶) from the hypothesis 𝐴𝐵𝐶𝐷𝐴′𝐵′𝐶′𝐷′ cube (1).
𝐴1 midpoint of segment |𝐵𝐴′|
(𝐴𝐵𝐴′) isosceles and 𝐴𝐴1 ⊥ 𝐵𝐴′
𝐴3 midpoint of |𝐴′𝐷|
} |𝐴1 𝐴3| mid-side in ∆𝐴′𝐵𝐷 ⟹ 𝐴1 𝐴3 ∥ 𝐵𝐷 (2)
From (1) and (2) ⟹ 𝐴1 𝐴3 ∥ (𝐴𝐴′𝐶) ⟹ 𝐴′𝐶 ⊥ 𝐴1 𝐴3 (3)

82
From ∆𝐴𝐶𝐴′:
From ∆𝐴𝐵𝐴′:
Similarly
In ∆𝐴𝐶𝐴′:
and
𝐴1 𝐴2 𝐴′ right with 𝑚(𝐴′𝐴2 𝐴1) = 90 because
From (4) and (3) ⟹ 𝐴′𝐶 ⊥ (𝐴1 𝐴2 𝐴3).
As
𝐴′𝐶 ⊥ (𝐴1 𝐴2 𝐴3)
𝐴′
𝐶 ⊥ 𝐴2 𝐴 (by construction)
} ⟹ 𝐴1 𝐴2 𝐴3 𝐴 coplanar ⟹ 𝐴1 𝐴2 𝐴3 𝐴 quadrilateral with
opposite angles 𝐴1 and 𝐴3 right ⟹ 𝐴1 𝐴2 𝐴3 𝐴 inscribable quadrilateral.
The conclusion is true only if ||𝐵𝐷|| = ||𝐴𝐶|| that is 𝑏 = 𝑐.
𝑀𝑁 ⊥ 𝐷𝐶 if

83
bisector plane
(𝑏 bisector half-plane)
In triangle 𝐷𝐷1 𝐷′:
But

84
From 1, 2, 3 ⟹
𝜎[𝐴𝐵𝐷]
𝜎[𝐴𝐵𝐶]
=
‖𝐷𝐸‖
‖𝐸𝐶‖
q.e.d.
Because the tetrahedron is regular 𝐴𝐵 = … =
we increase the denominator
If one of the points 𝑃 or 𝑄 is on face 𝐶𝐵𝐷 the problem is explicit.
We consider tetrahedron 𝑂𝑥𝑦𝑧, and prove that the sum of the measures of the
dihedral angles of this trihedron is bigger than 360°. Indeed: let 100′ be the internal
bisector of trihedron 𝑂𝑥𝑦𝑧 (1000′ the intersection of the bisector planes of the 3
dihedral angles) of the trihedron in 𝐴, 𝐵, 𝐶.

85
The size of each dihedron with edges 𝑜𝑥, 𝑜𝑦, 𝑜𝑧 is bigger than the size of the
corresponding angles of 𝐴𝐵𝐶, the sum of the measures of the dihedral angles of
trihedron 𝑂𝑥𝑦𝑧 is bigger than 180°.
Let (𝑎, 𝑏) be a plane ⊥ to 𝑜𝑧 at 𝐶; 𝑎 ⊥ 𝑜𝑧, 𝑏 ⊥ 𝑜𝑧, but |𝐶𝐴 and |𝐶𝐵 are on the same
half-space in relation to (𝑎𝑏) ⇒ 𝑚(𝐶̂) < 𝑚(𝑎𝑏̂).
In tetrahedron 𝐴𝐵𝐶𝐷, let 𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5 and 𝑎6 be the 6 dihedral angles formed
by the faces of the tetrahedron.
according to the inequality previously established.
We mark with a the intersection of planes (𝐴, 𝑑1) and (𝐵, 𝑑2). So
Let 𝑏 be a variable line that passes through 𝐶 and contained in 𝛼, which cuts 𝑑1
and 𝑑2 at 𝑀 respectively 𝑁. We have: 𝑀𝐴 ⊂ (𝐴, 𝑑1), 𝑀𝐴 ∩ 𝑁𝐵 = 𝑃(𝑀𝐴 and 𝑁𝐵
intersect because they are contained in the plane determined by (𝐴𝑀, 𝑏)).
Thus 𝑃 ∈ (𝐴, 𝑑1) and 𝑃 ∈ (𝐵, 𝑑2), ⟹ 𝑃 ∈ 𝑎, so 𝑃 describes line a the intersection of
planes (𝐴, 𝑑1) and (𝐵, 𝑑2).
Vice-versa: let 𝑄 ∈ 𝑎.
In the plane (𝐴, 𝑑1): 𝑄𝐴 ∩ 𝑑1 = 𝑀′
In the plane (𝐵, 𝑑2): 𝑄𝐵 ∩ 𝑑2 = 𝑁′

86
Lines 𝑁′𝑀′ and 𝐴𝐵 are coplanar (both are on plane (𝑄, 𝐴, 𝐵)). But because 𝑁′𝑀′ ⊂
𝛼 and 𝐴𝐵 has only point 𝐶 in common with 𝛼 ⟹ 𝑀′𝑁′ ∩ 𝐴𝐵 = 𝐶.
So 𝑀′𝑁′ passes through 𝐶. If planes (𝐴, 𝑑1) and (𝐵, 𝑑2) are parallel, the locus is the
empty set.
Remember the theorem: If a plane 𝛾 intersects two planes 𝛼 and 𝛽 such that
𝜎||𝛼 ⟹ (𝛾 ∩ 𝛼)||(𝛾 ∩ 𝛽). If plane (𝐿𝑀𝑁𝑃)||𝐵𝐷 we have:
If (𝐿𝑀𝑁𝑃)||𝐴𝐶 we have:
⟹ relation 𝑎.
Solution:
Let 𝐴′, 𝐵′, 𝐶′, 𝐷′ the projections of points 𝐴, 𝐵, 𝐶, 𝐷 onto plane (𝑀𝑁𝑃𝐿).
For ex. points 𝐵′, 𝐿, 𝐴′ are collinear on plane (𝐿𝑃𝑀𝑁) because they are on the
projection of line 𝐴𝐵 onto this plane.
Similarly we obtain:

87
By multiplying the 4 relations,
⟹ relation (𝑎) from 𝑑.
𝑀 – Midpoint of |𝐵𝐶|.
(1) 𝐴𝐵 ⊥ 𝐶𝐷 (hypothesis)
From 1 and 2
height in ∆𝐴𝐵𝐶 a
From 3 and 4 𝐴𝐶 ⊥ (𝐵𝐷𝐻) ⟹ 𝐴𝐶 ⊥ 𝐵𝐻 ⟹ 𝐵𝐻 height in ∆𝐴𝐵𝐶 b
From a and b ⟹ 𝐻 orthocenter ∆𝐴𝐵𝐶. Let 𝐶1 be the diametrical opposite point to
𝐶 in circle 𝐶(𝐴𝐵𝐶)𝐶𝐶1 diameter 𝑚(𝐶1 𝐵𝐶̂ ) = 900
but 𝐴𝐻 ⊥ 𝐵𝐶 ⟹ 𝐴𝐻 ∥ 𝐵𝐶1. Similarly
𝐵𝐻||𝐶1 𝐴, so 𝐴𝐻𝐵𝐶1 parallelogram, we have:

88
similarly
diametrical opposite to B
but
by substituting above, we have:
Let 𝑁, 𝑀, 𝑄, 𝑃, 𝑆, 𝑅 midpoints of the edges of the quadrilateral 𝑁𝑀𝑃𝑄 because:
𝑁𝑀||𝐶𝐷||𝑃𝑄 (median lines), 𝑄𝑀||𝐴𝐵||𝑃𝑁 (median lines), but 𝐶𝐷 ⊥ 𝐴𝐵 ⇒ 𝑀𝑁𝑄𝑃
rectangle |𝑁𝑄| ∩ |𝑃𝑀| = {0}.
Similarly 𝑀𝑆𝑃𝑅 rectangle with |𝑀𝑃| common diagonal with a, the first rectangle,
so the 6 points are equally distant from “𝑂” the midpoint of diagonals in the two
rectangles ⟹ the 6 points are on a sphere.
𝑀 arbitrary point on |𝐴𝐵′|
𝐴ʺ midpoint of segment [𝐵′𝐶′]
When 𝑀 = 𝐵′, point 𝑃 is in the position 𝐵′.

89
When 𝑀 = 𝐴, point 𝑃 is in the position {𝑃1} = [𝐴′𝐴1] ∩ [𝐴𝐴′′]
(𝐴′𝐴′′𝐴1 𝐴 rectangle, so 𝑃1 is the intersection of the diagonals of the rectangle)
[The locus is [𝐵′𝑃1]].
Let 𝑀 be arbitrary point 𝑀 ∈ |𝐴𝐵′|.
because:
By the way it was constructed
⟹ (∀) plane that contains 𝐴𝐴1 is perpendicular to (𝐵′𝐶𝐶′′), particularly to (𝐵′𝐴𝐴1) ⊥
(𝐵′𝐶′𝐶).
because 𝐵′, 𝑃 ∈ (𝐵′𝐴𝐴′′)
from this reason 𝐵′, 𝑃1 ∈ (𝐴′𝐵′𝐴1).
From 1 and 2
Let
So (∀) 𝑀 ∈ |𝐵′𝐴|
and we have
Vice-versa. Let 𝑃 arbitrary point, 𝑃 ∈ |𝐵′𝑃1| and
In plane
In plane
Indeed: 𝐴′𝐴′′ || (𝐵′𝐴𝐴1) thus any plane which passes through 𝐴′𝐴′′ will intersect
(𝐵′𝐴𝐴1) after a parallel line to 𝐴′𝐴′′. Deci 𝑀𝑁||𝐴′𝐴′′ or 𝑀𝑁||𝐴𝐴1 as 𝑀 ∈ (𝐵′𝐴𝐴1) ⟹
𝑀𝑁 ⊥ (𝐵′𝐶𝐶′′).
We’ve proved

90
and
we have
describes |𝐵′𝑃1| and vice-versa,
there is 𝑀|𝐵′𝐴| and 𝑁|𝐵′𝐴1| such that
and 𝑃 is the intersection of the diagonals of the quadrilateral 𝐴′𝑁𝑀𝐴′′.
known result
From 1 and 3
From the hypothesis:
𝑂𝐴 ⊥ 𝑂𝐵 ⊥ 𝑂𝐶 ⊥ 𝑂𝐴
We assume the problem is solved.
Let 𝑀 be the orthocenter of triangle 𝐴𝐵𝐶.

91
But 𝐶𝐶′ ⊥ 𝐴𝐵 ⟹
𝐴𝐵 ⊥ (𝑂𝐶𝐶′)
𝑀𝑂 ⊥ (𝐶𝑂𝐶′)
} ⟹ 𝐴𝐵 ⊥ 𝑂𝑀 ⟹ 𝑀𝑂 ⊥ 𝐴𝐵 (1)
𝐴𝑂 ⊥ (𝐶𝑂𝐵) ⟹ 𝐴𝑂 ⊥ 𝐵𝐶, but 𝐴𝐴′ ⊥ 𝐵𝐶 ⟹
𝐵𝐶 ⊥ (𝐴𝑂𝐴′)
𝑀𝑂 ⊂ (𝐴𝑂𝐴′)
} ⟹ 𝐵𝐶 ⊥ 𝑀𝑂 ⟹ 𝑀𝑂 ⊥ 𝐵𝐶 (2)
From (1) and (2) ⟹ 𝑀𝑂 ⊥ (𝐴𝐵𝐶)
So the plane (𝐴𝐵𝐶) that needs to be drawn must be perpendicular to 𝑂𝑀 at 𝑀.
𝐴′𝑁 ⊥ 𝐴𝐷, 𝐵′𝑀 ⊥ 𝐵𝐶
‖𝐵𝑀‖ =
𝑎−𝑎′
2
, ‖𝑃𝑀‖ =
𝑏−𝑏′
2
𝑣[𝐵𝑀𝑃𝑆𝐵′] =
𝑎 − 𝑎′
2
∙
𝑏 − 𝑏′
2
∙
ℎ
3
𝑣[𝑆𝑃𝑊𝑅𝐴′𝐵′] =
𝜎[𝑆𝑃𝐵′] ∙ ‖𝐵′𝐴′‖
3
=
𝑎 − 𝑎′
2
∙
ℎ
2
∙ 𝑏′
𝑣[𝐵′𝐴′𝑁𝑀𝐶′𝐷′𝐷1 𝐶1] =
(𝑏 + 𝑏′)ℎ
2
∙ 𝑎′
𝑣[𝐴𝐵𝐴′𝐵′𝐶𝐷𝐶′𝐷′] = 2 [2 ∙
𝑎−𝑎′
2
∙
𝑏−𝑏′
2
∙
ℎ
3
+
𝑎−𝑎′
2
∙
ℎ
2
∙ 𝑏′] + (
(𝑏+𝑏′)ℎ
2
) ∙ 𝑎′
=
ℎ
6
(2𝑎𝑏 − 2𝑎𝑏′
− 2𝑎′
𝑏′
+ 3𝑎𝑏′
− 3𝑎′
𝑏′
+ 3𝑎′
𝑏 + 3𝑎′
𝑏′) =
ℎ
6
[𝑎𝑏 + 𝑎′
𝑏′
+
(𝑎 + 𝑎′)(𝑏 + 𝑏′)].

92
𝑣[𝑂𝐴𝐵𝐵′] =
𝐵 ∙ ℎ
2
𝑣[𝑂𝐴′𝐵′𝐴] = 𝑣[𝐴𝐵𝑂𝑂′𝐴′𝐵′] − 𝑣[𝐴𝐵𝐵′
𝑂] − 𝑣[𝐴′
𝐵′
𝑂′
𝑂] =
ℎ
3
(𝐵 + 𝑏 + √𝐵𝑏) −
𝐵ℎ
3
−
𝑏ℎ
3
=
ℎ
3
√𝑏𝐵.
So:
𝑣[𝑂𝐴′𝐵′𝐴]
𝑣[𝐷𝐴𝐵𝐵′]
=
ℎ
3 √𝐵𝑏 ∙ √𝐵𝑏
𝐵ℎ
3 ∙ 𝐵
=
√𝑏
√𝐵
⟹ 𝑣[𝑂𝐴′𝐵′𝐴] =
√𝑏
√𝐵
∙ 𝑣[𝑂𝐴𝐵𝐵′]
For the relation above, determine the formula of the volume of the pyramid
frustum.
𝑑(𝐺𝐺′) = ℎ = 2𝑅
Let 𝑙 = ‖𝐴𝐶‖ ⟹ ‖𝐴𝐷‖ =
𝑙√3
2
⟹ ‖𝐺𝐷‖ =
𝑙√3
6
Figure 𝐺𝐷𝑀𝑂 rectangle ⟹ ‖𝐺𝐷‖ = ‖𝑂𝑀‖ ⟹
𝑙√3
6
= 𝑅 = 2√3𝑅

93
So, the lateral area is 𝑆𝑙 = 3 ∙ 2√3𝑅 ∙ 𝑅 = 12√3𝑅2
.
𝑣[𝐴𝐵𝐶𝐴′𝐵′𝐶′] = 𝜎[𝐴𝐵𝐶] ∙ 2𝑅 = 2√3𝑅 ∙
2√3𝑅√3
4
∙ 2𝑅 = 6√3𝑅2
.
The total area:
𝑆𝑡 = 𝑆𝑙 + 2𝜎[𝐴𝐵𝐶] = 12√3𝑅2
+ 2 ∙ 3𝑅√3𝑅2
= 18√3𝑅2
Let 𝑉1 and 𝑆1 be the volume, respectively the area obtained revolving around 𝑎.
𝑉2 and 𝑆2 be the volume, respectively the area obtained after revolving around 𝑏.
𝑉3 and 𝑆3 be the volume, respectively the area obtained after revolving around c.
So:
𝑉1 =
𝜋 ∙ 𝑖2(‖𝐶𝐷‖ + ‖𝐷𝐵‖)
3
=
𝜋 ∙ 𝑖2
∙ 𝑎
3
𝑆1 = 𝜋 ∙ 𝑖 ∙ 𝑐 + 𝜋 ∙ 𝑖 ∙ 𝑏 = 𝜋 ∙ 𝑖 ∙ (𝑏 + 𝑐)
𝑉2 =
𝜋𝑐2
𝑏
3
=
𝜋𝑐2
𝑏2
3𝑏
=
𝜋𝑏2
𝑐2
𝑎
3𝑎2
𝑆2 = 𝜋 ∙ 𝑐 ∙ 𝑎
𝑉3 =
𝜋𝑏2
𝑐
3
=
𝜋𝑏2
𝑐2
3𝑐
=
𝜋𝑏2
𝑐2
3𝑎
𝑆3 = 𝜋 ∙ 𝑏 ∙ 𝑎
Therefore:
1
𝑉1
2 =
1
𝑉2
2 +
1
𝑉3
2 ⟺
9𝑎2
(𝜋𝑏2 𝑐2)2
=
9𝑐2
(𝜋𝑏2 𝑐2)2
+
9𝑐2
(𝜋𝑏2 𝑐2)2
𝑆2
𝑆3
+
𝑆3
𝑆2
=
𝑆2 + 𝑆3
𝑆1
⟺
𝑐
𝑏
+
𝑏
𝑐
=
𝜋𝑎(𝑏 + 𝑐)
𝜋𝑖(𝑏 + 𝑐)
⟺
𝑐2
+ 𝑏2
𝑏 ∙ 𝑐
=
𝑎
𝑖
But 𝑖 ∙ 𝑎 = 𝑏 ∙ 𝑐 ⟹
𝑐2+𝑏2
𝑏𝑐
=
𝑎2
𝑏𝑐
, ‖𝐴𝐷‖ = 𝑖.

94
𝑟 = ‖𝑂𝐴‖ = 25 𝑐𝑚
𝑅 = ‖𝑂′𝐵‖ = 50 𝑐𝑚
2𝜋𝑟 = 1,57 ⟹ 𝑟 = 0,25 𝑚
2𝜋𝑅 = 3,14 ⟹ 𝑅 = 0,50 𝑚
‖𝐶𝑁‖ = 18 𝑐𝑚 = 0,18 𝑚
‖𝐴′𝐵‖ = 25 𝑐𝑚
‖𝐴𝐵‖ = √100 + 0,0625 = 10,003125
‖𝐴′𝑀‖ =
‖𝐴𝐴′‖ ∙ ‖𝐴′𝐵‖
‖𝐴𝐵‖
=
10 ∙ 0,25
10,003125
≈ 0,25
‖𝐶𝑁‖
‖𝐴′𝑀‖
=
‖𝐶𝐵‖
‖𝐵𝐴′‖
⟹
0,18
0,25
=
‖𝐶𝐵‖
0,25
⟹ ‖𝐶𝐵‖ = 0,18
‖𝑂′𝐶‖ = 𝑅′
= 0,50 − 0,18 = 0,32
‖𝑂𝑃‖ = 𝑟′
= 0,25 − 0,18 = 0,07
𝑉 =
𝜋1
3
(𝑅2
+ 𝑟2
+ 𝑅𝑟)
𝑉 =
𝜋10
3
(0,502
+ 0,252
+ 0,50 ∙ 0,25 − 0,322
− 0,072
− 0,32 ∙ 0,07)
=
𝜋10
3
(0,4375 − 0,1297) = 1,026𝜋𝑚3
‖𝑉𝑃‖ =
𝑎
2 sin
𝛼
2
cos
𝛼
2
In ∆𝑉𝐴𝑃: ‖𝑉𝐴‖ =
𝐴
2 sin
𝛼
2
.

95
In ∆𝑉𝐴𝑂′: ‖𝑉𝑂′‖2
=
𝑎2
4 sin2 𝛼
2
−
𝑎2
2
.
‖𝑉𝑂′‖ =
𝑎√cos 𝛼
2 sin
𝛼
2
In ∆𝑉𝑂𝑂′: ‖𝑂𝑂′‖ =
𝑎√cos 𝛼
2 sin
𝛼
2
− 𝑅.
𝑅2
=
𝑎2
2
+ (
𝑎√cos 𝛼
2 sin
𝛼
2
− 𝑅)
2
⟹
𝑎2
2
+
𝑎2
cos 𝛼
4 sin2 𝛼
2
−
2𝑎𝑅√cos 𝛼
2 sin
𝛼
2
⟹ 𝑎 =
4𝑅√cos 𝛼 sin2 𝛼
2
sin
𝛼
2 (2 cos2 𝛼
2 + cos 𝛼)
⟹ 𝑎 = 4𝑅√cos 𝛼 ∙ sin
𝛼
2
𝐴𝑙 = 4
𝑎2
cos
𝛼
2
2 ∙ 2 sin
𝛼
2
=
𝑎2
cos
𝛼
2
sin
𝛼
2
= 16𝑅2
cos 𝛼 sin2
𝛼
2
∙
cos
𝛼
2
sin
𝛼
2
= 8𝑅2
cos 𝛼 sin 𝛼 = 4𝑅2
sin 2𝛼
𝐴 𝑡 = 𝐴𝑙 + 𝑎2
= 4𝑅2
sin 2𝛼 + 16𝑅2
cos 𝛼 sin2
𝛼
2
‖𝑉𝑂′‖ = 𝑅 ⟹ 𝑅 =
𝑎√cos 𝛼
2 sin
𝑎
2
⟹ 𝑅 = 4𝑅√cos 𝛼 sin
𝛼
2
∙
√cos 𝛼
2 sin
𝛼
2
⟹ 2 cos 𝛼 = 1 ⟹ 𝛼 = 600
.

96
Various Problems
104. Determine the set of points in the plane, with affine coordinates 𝑧 that
satisfy:
a. |𝑧| = 1;
b. 𝜋 < arg 𝑧 ≤
3𝜋
2
; 𝑧 ≠ 0;
c. arg 𝑧 >
4𝜋
3
, 𝑧 ≠ 0;
d. |𝑧 + 𝑖| ≤ 2 .
105. Prove that the 𝑛 roots of the unit are equal to the power of the particular
root 𝜀1.
106. Knowing that complex number 𝑧 verifies the equation 𝑧 𝑛
= 𝑛, show that
numbers 2, −𝑖𝑧 and 𝑖𝑧 verify this equation.
Application: Find (1 − 2𝑖)4
and deduct the roots of order 4 of the number
−7 + 24𝑖.
107. Show that if natural numbers 𝑚 and 𝑛 are coprime, then the equations
𝑧 𝑚
− 1 = 0 and 𝑧 𝑛
− 1 = 0 have a single common root.
108. Solve the following binomial equation: (2 − 3𝑖)𝑧6
+ 1 + 5𝑖 = 0.
109. Solve the equations:

97
110. Solve the equation 𝑧̅ = 𝑧 𝑛−1
, 𝑛 ∈ 𝑁, where 𝑧̅ the conjugate of 𝑧.
111. The midpoints of the sides of a quadrilateral are the vertices of a
parallelogram.
112. Let 𝑀1 𝑀2 𝑀3 𝑀4 and 𝑁1 𝑁2 𝑁3 𝑁4 two parallelograms and 𝑃𝑖 the midpoints of
segments [𝑀𝑖 𝑁𝑖], 𝑖 ∈ {1, 2, 3, 4}. Show that 𝑃1 𝑃2 𝑃3 𝑃4 is a parallelogram or a
degenerate parallelogram.
113. Let the function 𝑓: 𝐶 → 𝐶, 𝑓(𝑧) = 𝑎𝑧 + 𝑏; (𝑎, 𝑏, 𝑐 ∈ 𝐶, 𝑎 ≠ 0). If 𝑀1 and
𝑀2 are of affixes 𝑧1 and 𝑧2, and 𝑀1
′
and 𝑀2
′
are of affixes 𝑓(𝑧1), 𝑓(𝑧2), show
that ‖𝑀1
′
𝑀2
′ ‖ = |𝑎| ∙ ‖𝑀1 𝑀2‖. We have ‖𝑀1
′
𝑀2
′ ‖ = ‖𝑀1 𝑀2‖ ⇔ |𝑎| = 1.
114. Prove that the function z → z̅, z ∈ C defines an isometry.
115. Let M1M2 be of affixes 𝑧1, 𝑧2 ≠ 0 and z2 = αz1. Show that rays |OM1, |OM2
coincide (respectively are opposed) ⟺ α > 0 (respectively α < 0).

98
116. Consider the points M1M2M3 of affixes 𝑧1 𝑧2 𝑧3 and 𝑀1 ≠ 𝑀2. Show that:
a. M3 ∈ |M1M2 ⟺
z3−z1
z2−z1
> 0;
b. M3 ∈ M1M2 ⟺
z3−z1
z2−z1
∈ R .
117. Prove Pompeiu’s theorem. If the point 𝑀 from the plane of the equilateral
triangle 𝑀1 𝑀2 𝑀3 ∉ the circumscribed circle ∆ 𝑀1 𝑀2 𝑀3  there exists a
triangle having sides of length ‖𝑀𝑀1‖, ‖𝑀𝑀2‖, ‖𝑀𝑀3‖.

99
Solutions
a.
|𝑧| = 1
|𝑧| = √𝑥2 + 𝑦2
} ⟹ 𝑥2
+ 𝑦2
= 1, so the desired set is the circle 𝐶(0,1).
b. 𝜋 < arg𝑧 ≤
3𝜋
2
.
The desired set is given by all the points of quadrant III, to which ray |𝑂𝑦 is added,
so all the points with 𝑥 < 0, 𝑦 < 0.
c.
arg𝑧 >
4𝜋
3
, 𝑧 ≠ 0
arg𝑧 ∈ [0, 2𝜋]
} ⟹
4𝜋
2
< arg𝑧 < 2𝜋
The desired set is that of the internal points of the angle with its sides positive
semi-axis and ray |𝑂𝐵.
d. |𝑧 + 𝑖| ≤ 2; 𝑧 = 𝑥 + 𝑦𝑖, its geometric image 𝑀.
where 𝑂′(0, −1).
Thus, the desired set is the disk centered at 𝑂(0,−1)
′
and radius 2.

100
Let the equation 𝑧4
= 𝑛. If 𝑧4
= 4 (𝑧 is the solution) then: (−𝑧)4
= (−1)4
𝑧4
= 1 ∙ 𝑛 =
𝑛, so – 𝑧 is also a solution.
⟹ 𝑖𝑧 is the solution;
⟹ −𝑖𝑧 is the solution;
⟹ is the solution of the equation 𝑧4
= −7 + 24𝑖.
The solutions of this equation are:
but based on the first part, if 𝑧 − 1 − 2𝑖 is a root, then
are solutions of the given equation.
If there exist 𝑘 and 𝑘′ with 𝑧 𝑘 = 𝑧 𝑘′, then

101
because (𝑚, 𝑛) = 1. Because 𝑘′
< 𝑛, 𝑘 < 𝑚, we have 𝑘′ = 0, 𝑘 = 0.
Thus the common root is 𝑧0.
As:

102
From:
positive
The given equation becomes
We find the sum of the abscissa of the opposite points:
⟹ 𝑀𝑁𝑃𝑄 a parallelogram.
In the quadrilateral 𝑀1 𝑀3 𝑁3 𝑁1 by connecting the midpoints we obtain the
parallelogram 𝑂′
𝑃1 𝑂′′
𝑃3, with its diagonals intersecting at 𝑂, the midpoint of |𝑂′𝑂′′|
and thus |𝑃1 𝑂| ≡ |𝑂𝑃3|. (1)

103
In the quadrilateral 𝑀4 𝑀2 𝑁2 𝑁4 by connecting the midpoints of the sides we obtain
the parallelogram 𝑂′
𝑃2 𝑂′′
𝑃4 with its diagonals intersecting in 𝑂, the midpoint of
|𝑂′𝑂′′| and thus |𝑃2 𝑂| ≡ |𝑂𝑃4|. (2)
From (1) and (2) 𝑃1 𝑃2 𝑃3 𝑃4 a parallelogram.
If:
If:
Let 𝑀1 and 𝑀2 be of affixes 𝑧1 and 𝑧2. Their images through the given function
𝑀1
′
and 𝑀2
′
with affixes 𝑧̅1 and 𝑧̅2, so

104
From (1) and (2) ⟹ ‖𝑀1 𝑀2‖ = ‖𝑀1
′
𝑀2
′ ‖ or ‖𝑀1
′
𝑀2
′ ‖ = |𝑧̅2 𝑧̅1| = |√ 𝑧2 − 𝑧1| =
|𝑧2 − 𝑧1| = ‖𝑀1 𝑀2‖.
So 𝑓: 𝐶 → 𝐶, 𝑓(𝑧) = 𝑧̅ defines an isometry because it preserves the distance
between the points.
We know that the argument (𝑎𝑧1) = arg𝑧1 + arg𝑧 𝛼 − 2𝑘𝜋, where 𝑘 = 0 or 𝑘 = 1.
Because arg𝑧2 = arg(𝑎𝑧1), arg𝑧2 = arg𝑧1 + arg𝑧 𝛼 − 2𝑘𝜋.
a. We assume that
Vice versa,
b. Let |𝑂𝑀1 and |𝑂𝑀2 be opposed ⟹ arg𝑧2 = arg𝑧1 + 𝜋
∈ to the negative ray |𝑂𝑥′
⟹ 𝛼 < 0. Vice versa,
𝑘 = 0 or 𝑘 = 1 ⟹
arg𝑧2 = arg𝑧1 + 𝜋
or
arg𝑧2 = arg𝑧1 − 𝜋
} ⟹ |𝑂𝑀1 and |𝑂𝑀2 are opposed.

105
If n and 𝑛′ are the geometric images of complex numbers 𝑧 and 𝑧′, then the image
of the difference 𝑧– 𝑧′ is constructed on |OM1| and |𝑀′𝑀| as sides.
We assume that 𝑀3 ∈ |𝑀1 𝑀2
We construct the geometric image of 𝑧2– 𝑧1 . It is the fourth vertex of the
parallelogram 𝑂𝑀1 𝑀2 𝑄1. The geometric image of 𝑧3– 𝑧1 is 𝑄2, the fourth vertex of the
parallelogram 𝑂𝑀1 𝑀3 𝑄2.
𝑂𝑄1 ∥ 𝑀1 𝑀2
𝑂𝑄2 ∥ 𝑀1 𝑀3
𝑀1 𝑀2 𝑀3collinear
} ⟹ 𝑄1, 𝑄2, 𝑄3 collinear ⟹
Vice versa, we assume that
If 𝑀3 and 𝑀2 ∈ the opposite ray to 𝑂, then 𝑧3– 𝑧1 = 𝛼(𝑧2– 𝑧1) with 𝛼 < 0.
We repeat the reasoning from the previous point for the same case.

106
Thus, when 𝑀3 ∈ 𝑀1 𝑀2 𝑀3 + 𝑀2 we obtain for the respective ratio positive,
negative or having 𝑀3 = 𝑀1, so
z3−z1
z2−z1
∈ R.
The images of the roots of order 3 of the unit are the peaks of the equilateral
triangle.
But ɛ1 = ɛ2
2
, so if we write ɛ2 = 𝜀, then ɛ1 = ɛ2.
Thus 𝑀1(1), 𝑀2(𝜀), 𝑀3(𝜀2).
We use the equality:
adequate (∀)𝑧 ∈ ℂ.
But
Therefore,
By substitution:
but

107
thus
Therefore ‖𝑀𝑀1‖, ‖𝑀𝑀2‖, ‖𝑀𝑀3‖ sides of a ∆.
Then we use ‖𝑥| − |𝑦‖ ≤ |𝑥 − 𝑦| and obtain the other inequality.

108
Problems in Spatial Geometry
118. Show that if a line 𝑑 is not contained in plane 𝛼, then 𝑑 ∩ 𝛼 is ∅ or it is
formed of a single point.
119. Show that (∀) 𝛼, (∃) at least one point which is not situated in 𝛼.
120. The same; there are two lines with no point in common.
121. Show that if there is a line 𝑑 (∃) at least two planes that contain line 𝑑.
122. Consider lines 𝑑, 𝑑′, 𝑑′′, such that, taken two by two, to intersect. Show
that, in this case, the 3 lines have a common point and are located on the
same plane.
123. Let 𝐴, 𝐵, 𝐶 be three non-collinear points and 𝐷 a point located on the
plane (𝐴𝐵𝐶). Show that:
a. The points 𝐷, 𝐴, 𝐵 are not collinear, and neither are 𝐷, 𝐵, 𝐶; 𝐷, 𝐶, 𝐴.
b. The intersection of planes (𝐷𝐴𝐵), (𝐷𝐵𝐶), (𝐷𝐶𝐴) is formed of a single
point.

109
124. Using the notes from the previous exercise, take the points 𝐸, 𝐹, 𝐺 distinct
from 𝐴, 𝐵, 𝐶, 𝐷, such that 𝐸 ∈ 𝐴𝐷, 𝐹 ∈ 𝐵𝐷, 𝐺 ∈ 𝐶𝐷. Let 𝐵𝐶 ∩ 𝐹𝐺 = {𝑃}, 𝐺𝐸 ∩
𝐶𝐴 = {𝑄}, 𝐸𝐹 ∩ 𝐴𝐵 = {𝑅}. Show that 𝑃, 𝑄, 𝑅 are collinear (T. Desarques).
125. Consider the lines 𝑑 and 𝑑′ which are not located on the same plane and
the distinct points 𝐴, 𝐵, 𝐶 ∈ 𝑑 and 𝐷, 𝐸 ∈ 𝑑′. How many planes can we draw
such that each of them contains 3 non-collinear points of the given points?
Generalization.
126. Show that there exist infinite planes that contain a given line 𝑑.
127. Consider points 𝐴, 𝐵, 𝐶, 𝐷 which are not located on the same plane.
a. How many of the lines 𝐴𝐵, 𝐴𝐶, 𝐴𝐷, 𝐵𝐶, 𝐵𝐷, 𝐶𝐷 can be intersected by a
line that doesn’t pass through 𝐴, 𝐵, 𝐶, 𝐷?
b. Or by a plane that doesn’t pass through 𝐴, 𝐵, 𝐶, 𝐷?
128. The points 𝛼 and 𝛽 are given, 𝐴, 𝐵 ∈ 𝛼. Construct a point 𝑀 ∈ 𝛼 at an
equal distance from 𝐴 and 𝐵, that ∈ also to plan 𝛽.
129. Determine the intersection of three distinct planes 𝛼, 𝛽, 𝛾.
130. Given: plane 𝛼, lines 𝑑1, 𝑑2 and points 𝐴, 𝐵 ∉ 𝛼 ∪ 𝑑1 ∪ 𝑑2. Find a point 𝑀 ∈
𝛼 such that the lines 𝑀𝐴, 𝑀𝐵 intersect 𝑑1 and 𝑑2.

110
131. There are given the plane 𝛼, the line 𝑑 ∉ 𝛼, the points 𝐴, 𝐵 ∉ 𝛼 ∪ 𝑑, and
𝐶 ∈ 𝛼. Let 𝑀 ∈ 𝑑 and 𝐴′, 𝐵′ the points of intersection of the lines 𝑀𝐴, 𝑀𝐵
with plane 𝛼 (if they exist). Determine the point 𝑀 such that the points
𝐶, 𝐴′, 𝐵′ to be collinear.
132. If points 𝐴 and 𝐵 of an open half-space 𝜎, then [𝐴𝐵] ⊂ 𝜎. The property is
as well adherent for a closed half-space.
133. If point 𝐴 is not situated on plane 𝛼 and 𝐵 ∈ 𝛼 then |𝐵𝐴 ⊂ |𝛼𝐴.
134. Show that the intersection of a line 𝑑 with a half-space is either line 𝑑 or
a ray or an empty set.
135. Show that if a plane 𝛼 and the margin of a half-space 𝜎 are secant
planes, then the intersection 𝜎 ∩ 𝛼 is a half-plane.
136. The intersection of a plane 𝛼 with a half-space is either the plane 𝛼 or a
half-plane, or an empty set.
137. Let 𝐴, 𝐵, 𝐶, 𝐷 four non coplanar points and 𝛼 a plane that doesn’t pass
through one of the given points, but it passes trough a point of the line
|𝐴𝐵|. How many of the segments |𝐴𝐵|, |𝐴𝐶|, |𝐴𝐷|, |𝐵𝐶|, |𝐵𝐷|, |𝐶𝐷| can be
intersected by plane 𝛼?

111
138. Let 𝑑 be a line and 𝛼, 𝛽 two planes such that 𝑑 ∩ 𝛽 = ∅ and 𝛼 ∩ 𝛽 = ∅.
Show that if 𝐴 ∈ 𝑑 and 𝐵 ∈ 𝛼, then 𝑑 ⊂ |𝛽𝐴 and 𝛼 ⊂ |𝛽𝐵.
139. Let |𝛼𝐴 and |𝛽𝐵 be two half-spaces such that 𝛼 ≠ 𝛽 and |𝛼𝐴 ⊂ |𝛽𝐵 or
|𝛼𝐴 ∩ |𝛽𝐵 = ∅. Show that 𝛼 ∩ 𝛽 = ∅.
140. Show that the intersection of a dihedral angle with a plane 𝛼 can be: a
right angle, the union of two lines, a line, an empty set or a closed half-
plane and cannot be any other type of set.
141. Let 𝑑 be the edge of a proper dihedron ∠ 𝛼′
𝛽′
, 𝐴 ∈ 𝛼′
– 𝑑, 𝑏 ∈ 𝛽′
– 𝑑 and
𝑃 ∈ int. ∠ 𝛼′
𝛽′
. Show that:
a. (𝑃𝑑) ∩ int. ∠ 𝛼′
𝛽′
= |𝑑𝑃;
b. If 𝑀 ∈ 𝑑, int. ∠ 𝐴𝑀𝐵 = int. 𝛼′
𝛽′
∩ (𝐴𝑀𝐵).
142. Consider the notes from the previous problem. Show that:
a. The points 𝐴 and 𝐵 are on different sides of the plane (𝑃𝑑);
b. The segment |𝐴𝐵| and the half-plane |𝑑𝑃 have a common point.
143. If ∠ 𝑎𝑏𝑐 is a trihedral angle, 𝑃 ∈ int. ∠ 𝑎𝑏𝑐 and 𝐴, 𝐵, 𝐶 are points on edges
𝑎, 𝑏, 𝑐, different from 𝑂, then the ray |𝑂𝑃 and int. 𝐴𝐵𝐶 have a common
point.
144. Show that any intersection of convex sets is a convex set.

112
145. Show that the following sets are convex planes, half-planes, any open or
closed half-space and the interior of a dihedral angle.
146. Can a dihedral angle be a convex set?
147. Which of the following sets are convex:
a. a trihedral angle;
b. its interior;
c. the union of its faces;
d. the union of its interior with all its faces?
148. Let 𝜎 be an open half-space bordered by plane 𝛼 and 𝑀 a closed convex
set in plane 𝛼. Show that the set 𝑀 ∩ 𝜎 is convex.
149. Show that the intersection of sphere 𝑆(𝑂, 𝑟) with a plane which passes
through 𝑂, is a circle.
150. Prove that the int. 𝑆(𝑂, 𝑟) is a convex set.
151. Show that, by unifying the midpoints of the opposite edges of a
tetrahedron, we obtain concurrent lines.

113
152. Show that the lines connecting the vertices of a tetrahedron with the
centroids of the opposite sides are concurrent in the same point as the
three lines from the previous example.
153. Let 𝐴𝐵𝐶𝐷 be a tetrahedron. We consider the trihedral angles which have
as edges [𝐴𝐵, [𝐴𝐸, [𝐴𝐷, [𝐵𝐴, [𝐵𝐶, [𝐵𝐷, [𝐶𝐴, [𝐶𝐵, [𝐶𝐷, [𝐷𝐴, [𝐷𝐵, [𝐷𝐶. Show that
the intersection of the interiors of these 4 trihedral angles coincides with
the interior of tetrahedron [𝐴𝐵𝐶𝐷].
154. Show that (∀) 𝑀 ∈ int. [𝐴𝐵𝐶𝐷] (∃) 𝑃 ∈ |𝐴𝐵| and 𝑄 ∈ |𝐶𝐷| such that 𝑀 ∈
‖𝑃𝑄.
155. The interior of tetrahedron [𝐴𝐵𝐶𝐷] coincides with the union of segments
|𝑃𝑄| with 𝑃 ∈ |𝐴𝐵| and 𝑄 ∈ |𝐶𝐷|, and tetrahedron [𝐴𝐵𝐶𝐷] is equal to the
union of the closed segments [𝑃𝑄], when 𝑃 ∈ [𝐴𝐵] and 𝑄 ∈ [𝐶𝐷].
156. The tetrahedron is a convex set.
157. Let 𝑀1 and 𝑀2 convex sets. Show that by connecting segments [𝑃𝑄], for
which 𝑃 ∈ 𝑀1 and 𝑄 ∈ 𝑀2 we obtain a convex set.
158. Show that the interior of a tetrahedron coincides with the intersection of
the open half-spaces determined by the planes of the faces and the
opposite peak. Define the tetrahedron as an intersection of half-spaces.

114
Solutions
We assume that 𝑑 ∩ 𝛼 = {𝐴, 𝐵} ⇒ 𝑑 ⊂ 𝛼.
It contradicts the hypothesis ⟹ 𝑑 ∩ 𝛼 = {𝐴} or 𝑑 ∩ 𝛼 = ∅.
We assume that all the points belong to the plane 𝛼 ⟹ (∄) for the points that
are not situated in the same plane. False!
∃ 𝐴, 𝐵, 𝐶, 𝐷, which are not in the same plane. We assume that 𝐴𝐵 ∩ 𝐶𝐷 = {0} ⟹
𝐴𝐵 and 𝐶𝐷 are contained in the same plane and thus 𝐴, 𝐵, 𝐶, 𝐷 are in the same
plane. False, it contradicts the hypothesis ⟹ 𝐴𝐵 ∩ 𝐶𝐷 = ∅ ⟹ (∃) lines with no
point in common.
(∃) 𝐴 ∉ 𝑑 (if all the points would ∈ 𝑑, the existence of the plane and space would
be negated). Let 𝛼 = (𝑑𝐴), (∃)𝐵 ∉ 𝛼 (otherwise the space wouldn’t exist). Let 𝛽 =
(𝐵𝑑), 𝛼 ≠ 𝛽 and both contain line 𝑑.
We show that 𝑑 ≠ 𝑑′
≠ 𝑑′′
≠ 𝑑.
Let 𝑑 ∩ 𝑑′
= {𝐴} = (𝑑, 𝑑′) ⟹ {
𝑑 ⊂ 𝛼
𝑑′
⊂ 𝛼
𝑑 ∩ 𝑑′
= {𝐵}
𝐵 ≠ 𝐴
⟹ {
𝐵 ∈ 𝑑
𝑑 ⊂∝
⟹ 𝐵 ∈ 𝑎 , 𝐵 ∈ 𝑑′

115
𝑑′′
∩ 𝑑′
= {𝐶}
𝐶 ≠ 𝐵
𝐶 ≠ 𝐴
⟹ 𝐶 ∈ 𝑑′
𝑑′
⊂ ∝
⟹ 𝐶 ∈ 𝛼, 𝐶 ∈ 𝑑′′
⟹
𝑑 = 𝑑′
or
𝑑 = 𝑑′′
⟹ 𝑑′′
⊂ 𝛼, so the lines are located on the same plane α.
If
𝑑 ∩ 𝑑′
= {𝐴} ⟹ 𝐴 ∈ 𝑑′
𝑑′′
∩ 𝑑 = {𝐴} ⟹ 𝐴 ∈ 𝑑′′} ⟹ 𝑑′
∩ 𝑑′′
= {𝐴}, and the three lines have a point in
common.
a. 𝐷 ∉ (𝐴𝐵𝐶).
We assume that 𝐷, 𝐴, 𝐵 collinear ⟹ (∃)𝑑 such that
𝐷 ∈ 𝑑, 𝐴 ∈ 𝑑, 𝐵 ∈ 𝑑
𝐴 ∈ (𝐴𝐵𝐶), 𝐵 ∈ (𝐴𝐵𝐶)
}
𝑇2
⇒ 𝑑 ⊂
(𝐴𝐵𝐶) ⟹ 𝐷 ∈ (𝐴𝐵𝐶) – false. Therefore, the points 𝐷, 𝐴, 𝐵 are not collinear.
b. Let (𝐷𝐴𝐵) ∩ (𝐵𝐶𝐷) ∩ (𝐷𝐶𝐴) = 𝐸.
As the planes are distinct, their intersections are:

116
(𝐷𝐴𝐵) ∩ (𝐷𝐵𝐶) = 𝐷𝐵
(𝐷𝐴𝐵) ∩ (𝐷𝐶𝐴) = 𝐷𝐴
(𝐷𝐵𝐶) ∩ (𝐷𝐶𝐴) = 𝐷𝐶
}
⟹ If (𝐷𝐴𝐵) = (𝐷𝐵𝐶)
⟹ 𝐴, 𝐵, 𝐶, 𝐷 coplanar, contrary to the hypothesis.
We suppose that (∃)𝑀 ∈ 𝐸, 𝑀 ≠ 𝐷 ⟹
𝑀 ∈ 𝐷𝐵
𝑀 ∈ 𝐷𝐴
} ⟹
𝐵 ∈ 𝑀𝐷
𝐴 ∈ 𝑀𝐷
} ⟹ 𝐴, 𝐵, 𝐷 are collinear
(false, contrary to point a.). Therefore, set 𝐸 has a single point 𝐸 = {𝐷}.
We showed at the previous exercise that if 𝐷 ∉ (𝐴𝐵𝐶), (𝐷𝐴𝐵) ≠ (𝐷𝐵𝐶). We show
that 𝐸, 𝐹, 𝐺 are not collinear. We assume the opposite. Then,
Having three common points 𝐷, 𝐵 and 𝐺 ⟹ false. So 𝐸, 𝐹, 𝐺 are not collinear and
determine a plane (𝐸𝐹𝐺).
⟹ 𝑃, 𝑄, 𝑅 are collinear because ∈ to the line of intersection of the two planes.

117
The planes are (𝐴, 𝑑′); (𝐵, 𝑑′); (𝐶, 𝑑′).
Generalization: The number of planes corresponds to the number of points on
line 𝑑 because 𝑑′ contains only 2 points.
Let line 𝑑 be given, and 𝐴 any point such that 𝐴 ∉ 𝑑.
We obtain the plane 𝛼 = (𝐴, 𝑑), and let 𝑀 ∉ 𝛼. The line 𝑑′ = 𝐴𝑀, 𝑑′ ⊄ 𝛼 is not
thus contained in the same plane with 𝑑. The desired planes are those of type
(𝑀𝑑), 𝑀 ∈ 𝑑′, that is an infinity of planes.
a. (∀) 3 points determine a plane. Let plane (𝐴𝐵𝐷). We choose in this plane 𝑃 ∈
|𝐴𝐷| and 𝑄 ∈ |𝐴𝐵| such that 𝑃 ∈ |𝐵𝑄|, then the line 𝑃𝑄 separates the points 𝐴
and 𝐷, but does not separate 𝐴 and 𝐵, so it separates 𝑃 and 𝐷 ⇒ 𝑃𝑄 ∩ |𝐵𝐷| = 𝑅,
where 𝑅 ∈ |𝐵𝐷|.
Thus, the line 𝑃𝑄 meets 3 of the given lines. Let’s see if it can meet more.

118
We assume that
it has two points in common with the plane.
⟹ 𝐴, 𝐵, 𝐶, 𝐷 coplanar – false.
Thus,
false.
We show in the same way that 𝑃𝑄 does not cut 𝐴𝐶 or 𝐷𝐶, so a line meets at
most three of the given lines.
b. We consider points 𝐸, 𝐹, 𝐺 such that 𝐸 ∈ |𝐵𝐶|, 𝐴 ∈ |𝐷𝐹|, 𝐷 ∈ |𝐵𝐺|. These points
determine plane (𝐸𝐹𝐺) which obviously cuts the lines 𝐵𝐶, 𝐵𝐷 and 𝐵𝐷. 𝐹𝐺 does not
separate 𝐴 and 𝐷 or 𝐵𝐷 ⟹ it does not separate 𝐴 or 𝐵 ⟹ 𝐴 ∈ |𝐵𝑅|.

119
Let’s show that (𝐸𝐹𝐺) meets as well the lines 𝐴𝐵, 𝐶𝐷, 𝐴𝐶. In the plane (𝐴𝐵𝐷) we
consider the triangle 𝐹𝐷𝐺 and the line 𝐴𝐵.
As this line cuts side |𝐹𝐷|, but it does not cut |𝐷𝐺|, it must cut side |𝐹𝐺|, so 𝐴𝐵 ∩
|𝐹𝐺| = {𝑅} ⟹ 𝑅 ∈ |𝐹𝐺| ⊂ (𝐸𝐹𝐺), so 𝐴𝐵 ∩ (𝐸𝐹𝐺) = {𝑅}. In the plane (𝐵𝐶𝐷), the line
𝐸𝐺 cuts |𝐵𝐶| and does not cut |𝐵𝐷|, so 𝐸𝐺 cuts the side |CD|, 𝐸𝐺 ∩ |𝐶𝐷| = {𝑃} ⟹
𝑃 ∈ 𝐸𝐺 ⊂ (𝐸𝐹𝐺) ⟹ 𝐶𝐷 ∩ (𝐸𝐹𝐺) = {𝑃}.
𝑅 ∈ (𝐸𝐹𝐺), 𝑅 does not separate 𝐴 and 𝐵
𝐸 separates 𝐵 and 𝐶
} ⟹ 𝑅 ∈∩ |𝐴𝐶| = 𝑄 ⟹ 𝑄 ∈ 𝑅𝐸 ⟹ 𝑄
∈ (𝐸𝐹𝐺) ∩ 𝐴𝐶 = {𝑄}.
We assume problem is solved, if
𝑀 ∈ 𝛼
𝑀 ∈ 𝛽
} ⟹ 𝛼 ∩ 𝛽 ≠ ∅, ⟹ 𝛼 ∩ 𝛽 = 𝑑.
As ||𝑀𝐴|| = ||𝑀𝐵|| ⟹ 𝑀 ∈ the bisecting line of the segment [𝐴𝐵].
So, to find 𝑀, we proceed as follows:
1. We look for the line of intersection of planes 𝛼 and 𝛽, d. If 𝛼 ∥ 𝛽, the
problem hasn’t got any solution.
2. We construct the bisecting line 𝑑′ of the segment [𝐴𝐵] in the plane 𝛼.
3. We look for the point of intersection of lines 𝑑 and 𝑑′. If 𝑑 ∥ 𝑑′, the problem
hasn’t got any solution.

120
If 𝛼 ∩ 𝛽 = ∅ ⟹ 𝛼 ∩ 𝛽 ∩ 𝛾 = ∅. If 𝛼 ∩ 𝛽 = 𝑑, the desired intersection is 𝑑 ∩ 𝛾, which
can be a point (the 3 planes are concurrent), the empty set (the line of intersection
of two planes is || with the third) or line 𝑑 (the 3 planes which pass through 𝑑 are
secant).
To determine 𝑀, we proceed as follows:
1. We construct plane (𝐴𝑑1) and we look for the line of intersection with 𝛼1, 𝑑1.
If 𝑑1 (/∃), ∄ neither does 𝑀.
2. We construct plane (𝐵𝑑2) and we look for the line of intersection with 𝛼, 𝑑2′.
If 𝑑2′ does not exist, neither does 𝑀.
3. We look for the point of intersection of lines 𝑑1′ and 𝑑2′. The problem has
only one solution if the lines are concurrent, an infinity if they are coinciding
lines and no solution if they are parallel.
We assume the problem is solved.
a. First we assume that 𝐴. 𝐵, 𝐶 are collinear. As 𝐴𝐴′ and 𝐵𝐵′ are concurrent lines,
they determine a plane 𝛽, that intersects 𝛼 after line 𝐴′𝐵′.
As
and points 𝐶, 𝐴, 𝐵′ are collinear (∀)𝑀 ∈ 𝑑.

121
b. We assume that 𝐴, 𝐵, 𝐶 are not collinear.
We notice that: (𝐴𝐴′, 𝐵𝐵′) = 𝛽 (plane determined by 2 concurrent lines).
𝛽 ∪ 𝛼 = 𝑑′ and 𝐶 ∈ 𝑑′.
To determine 𝑀 we proceed as follows:
1) We determine plane (𝐴𝐵𝐶);
2) We look for the point of intersection of this plane with line 𝑑, so 𝑑 ∩
(𝐴𝐵𝐶) = {𝑀} is the desired point.
Then (𝐴𝐵𝐶) ∩ 𝛼 = 𝑑′
.
⟹ 𝐴′, 𝐵′, 𝐶′ are collinear.
𝐴 ∈ 𝜎 and 𝐵 ∈ 𝜎 ⟹ [𝐴𝐵] ∩ 𝛼 ≠ ∅.
Let 𝜎 = |𝛼𝐴 = |𝛼𝐵.
Let 𝑀 ∈ |𝐴𝐵| and we must show that 𝑀 ∈ 𝜎(∀)𝑀 inside the segment.
We assume the contrary that 𝑀 ∉ 𝜎 ⟹ (∃)𝑃 such that [𝐴𝑀] ∩ 𝑑 = {𝑃} ⟹ 𝑃 ∈
[𝐴𝑀] ⟹ 𝑃 ∈ [𝐴𝐵] ⟹ [𝐴𝐵] ∩ 𝛼 ≠ ∅ false.
𝑃 ∈ 𝛼, so 𝑀 ∈ 𝜎. The property is also maintained for the closed half-space.

122
Compared to the previous case there can appear the situation when one of the
points 𝐴 and 𝐵 ∈ 𝛼 or when both belong to 𝛼.
If 𝐴 ∈ 𝛼, 𝐵 ∈ 𝜎, |𝐴𝐵| ∩ 𝛼 ≠ ∅ and we show as we did above that:
If:
Let
So
Let 𝛼 be a plane and 𝜎1, 𝜎2 the two half-spaces that it determines. We consider
half-space 𝜎1.

123
𝑃 determines on 𝑑 two rays, |𝑃𝐴 and |𝑃𝐵 where
𝐴 and 𝐵 are in different half-spaces.
We assume
Let 𝜎 be an open half-space and 𝑝 its margin and let 𝑑 = 𝛼 ∩ 𝛽.
We choose points 𝐴 and 𝐵 ∈ 𝛼 – 𝑑, on both sides of line 𝑑 ⟹
⟹ 𝐴, 𝐵 are on one side and on the other side of 𝛽 and it means that only one of
them is on 𝜎.
We assume that 𝐴 ∈ 𝜎 ⟹ 𝐵 ∈ 𝜎. We now prove 𝛼 ∩ 𝜎 = |𝑑𝐴.
𝛼 ∩ 𝜎 ⊂ |𝑑𝐴

124
Let
[𝑀𝐵] ∈ 𝑑 ≠ ∅ ⟹ 𝑀 and 𝐵 are on one side and on the other side of line 𝑑 ⟹ 𝑀 is
on the same side of line 𝑑 with 𝐴 ⟹ 𝑀 ∈ |𝑑𝐴
⟹ 𝑀 ∈ 𝑎 ∩ 𝜎, so |𝑑𝐴 ⊂ 𝑎 ∩ 𝜎.
Let σ be the considered half-space and β its margin. There are more possible
cases:
In this case it is possible that:
Let
is a half-plane according to a previous problem.

125
The intersection of two planes is a line and it cuts only two sides of a triangle.
There are more possible cases:
1. 𝑑 cuts |𝐴𝐵| and |𝐵𝐶|
𝑑′ cuts |𝐴𝐵| and |𝐴𝐷|, 𝛼 cuts |𝐴𝐷| so it has a point in common with (𝐴𝐷𝐶) and let
(𝐴𝐷𝐶) ∩ 𝛼 = 𝑑′′.
𝑑′′ cuts |𝐴𝐷| and does not cut |𝐴𝐶| ⟹ 𝑑′′ cuts |𝐷𝐶|
𝛼 cuts |𝐷𝐶| and |𝐵𝐶| ⟹ it does not cut |𝐵𝐷|. In this case 𝛼 cuts 4 of the 6
segments (the underlined ones).
2. 𝑑 cuts |𝐴𝐵| and |𝐴𝐶|, it does not cut |𝐵𝐷|
𝑑′ cuts |𝐴𝐵| an |𝐴𝐷|, it does not cut |𝐵𝐷|
𝑑′′ cuts |𝐴𝐷| an |𝐴𝐶|, it does not cut |𝐷𝐶|
⟹ 𝛼 does not intersect plane (𝐵𝐶𝐷). In this case 𝛼 intersects only 3 of the 6
segments.
3. 𝑑 cuts |𝐴𝐵| and |𝐵𝐶|, it does not cut |𝐴𝐶|
𝑑′ cuts |𝐴𝐵| an |𝐵𝐷|, it does not cut |𝐷𝐶|
𝛼 intersects |𝐵𝐷| and |𝐵𝐶|, so it does not cut |𝐷𝐶|
In ∆𝐵𝐷𝐶 ⟹ 𝛼 does not intersect plane (𝐴𝐷𝐶)
In this case 𝛼 intersects only three segments.
4. 𝑑 cuts |𝐴𝐵| and |𝐴𝐶|, it does not cut |𝐵𝐶|
𝑑′ cuts |𝐴𝐵| an |𝐵𝐷|, it does not cut |𝐴𝐷|
𝑑′′ cuts |𝐴𝐶| an |𝐷𝐶|
𝛼 does not cut |𝐵𝐶| in triangle 𝐵𝐷𝐶. So 𝛼 intersects 4 or 3 segments.
Let
Let

126
We first assume that 𝛼 ≠ 𝛽 and |𝛼𝐴 ⊂ |𝛽𝐵.
As
The hypothesis can then be written as 𝛼 ≠ 𝛽 and |𝛼𝐴 ⊂ |𝛽𝐵. Let’s show that 𝛼 ∩
𝛽 = ∅. By reductio ad absurdum, we assume that 𝛼 ∩ 𝛽 ≠ ∅ ⟹ (∃)𝑑 = 𝛼 ∩ 𝛽 and let
𝑂 ∈ 𝑑, so 𝑂 ∈ 𝛼 and 𝑂 ∈ 𝛽. We draw through 𝐴 and 𝑂 a plane 𝑟, such that 𝑑 ∈ 𝑟, so
the three planes 𝛼, 𝛽 and 𝑟 do not pass through this line. As 𝑟 has the common
point 𝑂 with 𝛼 and 𝛽, it is going to intersect these planes.
which is a common point of the 3 planes. Lines 𝛿 and 𝛿′ determine 4 angles in
plane 𝑟, having 𝑂 as a common peak, 𝐴 ∈ the interior of one of them, let 𝐴 ∈ int.
ℎ𝑘̂. We consider 𝐶 ∈ int. ℎ𝑘̂.
Then 𝐶 is on the same side with 𝐴 in relation to 𝛿′, so 𝐶 is on the same side with
𝐴 in relation to 𝛼 ⟹ 𝐶 ∈ |𝛼𝐴.

127
But 𝐶 is on the opposite side of 𝐴 in relation to 𝛿, so 𝐶 is on the opposite side of
𝐴 in relation to 𝛽 ⟹ 𝐶 ∉ |𝛽𝐴. So |𝛼𝐴 ⊄ |𝛽𝐴 – false – it contradicts the hypothesis ⟹
So 𝛼 ∩ 𝛽 = ∅.
Let 𝑑 be the edge of the given dihedral angle. Depending on the position of a line
in relation to a plane, there can be identified the following situations:
The ray with its origin in 𝑂, so 𝛼 ⊂ 𝛽′𝛾′̂ = 𝑑′𝑑′′̂ thus an angle.
Indeed, if we assumed that 𝑑′ ∩ 𝑑′′ ≠ ∅ ⟹ (∃)𝑂 ∈ 𝑑′ ∩ 𝑑′′.
false – it contradicts the hypothesis.

128
Or
in this case 𝛼 ∩ 𝛽′ 𝛾′̂ = 𝑑′′ - a line.
Then 𝛼 ∩ 𝛽′ 𝛾′̂ = ∅.
𝑑 ∩ 𝛼 = 𝑑, but 𝛼 ≠ 𝛽, 𝛼 ≠ 𝛾
𝛼 ∩ 𝛽′ 𝛾′̂ = 𝑑 thus the intersection is a line.
In this case the intersection is a closed half-plane.
is a half-plane

129
From (*) and (**),
⟹ points 𝐴 and 𝐵 are on different sides of (𝑑𝑃).
is a half-plane
so they are secant planes

130
Let rays
As 𝑃 is interior to the dihedron formed by any half-plane passing through 𝑂 of the
trihedral, so
So
𝑃 and 𝑄 in the same half-plane det. 𝑂𝐴 ⇒ 𝑃 and 𝑄 on the same side of 𝑂𝐴 (1)
⟹ 𝑃 and 𝐴 are on the same side of (𝑂𝐵𝐶) ∩ 𝛾′ ⟹ 𝑃 and 𝐴 are on the same side of
𝑂𝑄 (2).
From (1) and (2) ⟹
Let 𝑀 and 𝑀′′be two convex sets and 𝑀 ∩ 𝑀′ their intersection. Let
so the intersection is convex.
a. Let 𝑃, 𝑄 ∈ 𝛼; 𝑃 ≠ 𝑄 ⟹ |𝑃𝑄 = 𝑃𝑄 (the line is a convex set)
𝑃𝑄 ⊂ 𝛼, so |𝑃𝑄| ⊂ 𝛼, so the plane is a convex set.

131
b. Half-planes: Let 𝑆 = |𝑑𝐴 and 𝑃, 𝑄 ∈ 𝑆 ⟹ |𝑃𝑄| ∩ 𝑑 = ∅. Let 𝑀 ∈ |𝑃𝑄| ⟹ |𝑃𝑀| ⊂
|𝑃𝑄| ⟹ |𝑃𝑀| ∩ 𝑑 = ∅ ⟹ 𝑃 and 𝑀 are in the same half-plane ⟹ 𝑀 ∈ 𝑆. So
|𝑃𝑄| ⊂ 𝑆 and 𝑆 is a convex set.
Let 𝑆′ = [𝑑𝐴. There are three situations:
1) 𝑃, 𝑄 ∈ |𝑑𝐴 – previously discussed;
2) 𝑃, 𝑄 ∈ 𝑑 ⟹ |𝑃𝑄| ⊂ 𝑑 ⊂ 𝑆′;
3) 𝑃 ∈ 𝑑, 𝑄 ∉ 𝑑 ⟹ |𝑃𝑄| ⊂ |𝑑𝑄 ⟹ |𝑃𝑄| ⊂ |𝑑𝐴 ⊂ [𝑑𝐴 so [𝑑𝐴 is a convex set.
c. Half-spaces: Let 𝜎 = |𝛼𝐴 and let 𝑃, 𝑄 ∈ 𝜎 ⟹ |𝑃𝑄| ∩ 𝛼 = ∅.
Let 𝑀 ∈ |𝑃𝑄| ⟹ |𝑃𝑀| ⊂ |𝑃𝑄| ⟹ |𝑃𝑀| ∩ 𝛼 = ∅.
Let 𝜎′ = [𝛼𝐴. There are three situations:
1) 𝑃, 𝑄 ∈ |𝛼𝐴 previously discussed;
2) 𝑃, 𝑄 ∈ 𝛼 ⟹ |𝑃𝑄| ⊂ 𝛼 ⊂ 𝜎′;
3) 𝑃 ∈ 𝛼, 𝑄 ∉ 𝛼.
and so 𝜎′ is a convex set.
d. the interior of a dihedral angle:
𝑖𝑛𝑡. 𝛼′𝛽′ = |𝛼𝐴 ∩ |𝛽𝐵 and as each half-space is a convex set and their intersection
is the convex set.

132
No. The dihedral angle is not a convex set, because if we consider it as in the
previous figure 𝐴 ∈ 𝛽′ and 𝐵 ∈ 𝛼′.
Only in the case of the null or straight angle, when the dihedral angle
becomes a plane or closed half-plane, is a convex set.
a. No. The trihedral angle is not the convex set, because, if we take 𝐴 ∈ 𝑎 and 𝑄 ∈
the int. 𝑏𝑐̂ determined by 𝑃 ∈ the int. 𝑎𝑏𝑐̂ , (∃)𝑅 such that |𝑂𝑃 ∩ the int. 𝐴𝐵𝐶 =
{𝑅}, 𝑅 ∈ |𝐴𝑄|, 𝑅 ∉ 𝑎𝑏𝑐̂ .
So 𝐴, 𝑄 ∈ 𝑎𝑏𝑐̂ , but |𝐴𝑄| ∉ 𝑎𝑏𝑐.
b. 𝛣 = (𝑂𝐶𝐴), 𝛾 = (𝑂𝐴𝐵) is a convex set as an intersection of convex sets.
C) It is the same set from a. and it is not convex.
D) The respective set is [𝛼𝐴 ∩ [𝛽𝐵 ∩ [𝛾𝐶, intersection of convex sets and, thus, it
is convex.
Let 𝜎 = |𝛼𝐴 and 𝑀 ⊂ 𝛼. Let 𝑃, 𝑄 ∈ 𝑀 ∩ 𝜎.
We have the following situations:

133
Let
In plane (𝑂𝑃𝑄), let 𝑀 ∈ (𝑃𝑄).
Let: 𝑃 midpoint of |𝐴𝐵|
𝑅 midpoint of |𝐵𝐶|
𝑄 midpoint of |𝐷𝐶|
𝑆 midpoint of |𝐴𝐷|
𝑇 midpoint of |𝐵𝐷|
𝑈 midpoint of |𝐴𝐶|
or

134
In triangle ABC:
In triangle DAC:
⟹ parallelogram ⟹ |𝑃𝑄| and |𝑆𝑅| intersect at their midpoint 𝑂.
⟹ 𝑆𝑇𝑅𝑈 parallelogram.
⟹ |𝑇𝑈| passes through midpoint 𝑂 of |𝑆𝑅|.
Thus the three lines 𝑃𝑅, 𝑆𝑅, 𝑇𝑈 are concurrent in 𝑂.
Let tetrahedron 𝐴𝐵𝐶𝐷 and 𝐸 be the midpoint of |𝐶𝐷|. The centroid 𝐺 of the face
𝐴𝐶𝐷 is on |𝐴𝐸| at a third from the base. The centroid 𝐺′ of the face 𝐵𝐶𝐷 is on |𝐵𝐸|
at a third from the base |𝐶𝐷|.
We separately consider ∆𝐴𝐸𝐵. Let 𝐹 be the midpoint of 𝐴𝐵, so 𝐸𝐹 is median in
this triangle and, in the previous problem, it was one of the 3 concurrent segments
in a point located in the middle of each.
Let 𝑂 be the midpoint of |𝐸𝐹|. We write 𝐴𝑂 ∩ 𝐸𝐵 = {𝐺′} and 𝐵𝐺 ∩ 𝐸𝐴 = {𝐺}.

135
From (1) and (2)
⟹ 𝐺′ is exactly the centroid of face 𝐵𝐶𝐷, because it is situated on median |𝐸𝐵| at
a third from 𝐸. We show in the same way that 𝐺 is exactly the centroid of face 𝐴𝐶𝐷.
We’ve thus shown that 𝐵𝐺 and 𝐴𝐺′ pass through point 𝑂 from the previous
problem.
We choose faces 𝐴𝐶𝐷 and 𝐴𝐶𝐵 and mark by 𝐺′′ the centroid of face 𝐴𝐶𝐵, we
show in the same way that 𝐵𝐺 and 𝐷𝐺′′ pass through the middle of the segment
|𝑀𝑁| (|𝐴𝑀| ≡ |𝑀𝐶|, |𝐵𝑁| ≡ |𝑁𝐷|) thus also through point 𝑂, etc.
We mark planes (𝐴𝐵𝐶) = 𝛼, (𝐴𝐷𝐶) = 𝛽, (𝐵𝐷𝐶) = 𝛾, (𝐴𝐵𝑂) = 𝛿.
Let 𝑀 be the intersection of the interiors of the 4 trihedral angles.
We show that:
𝑀 = int. [𝐴𝐵𝐶𝐷], by double inclusion.
1. 𝑃 ∈ 𝑀 ⟹ 𝑃 ∈ int. 𝑎𝑏𝑐̂ ∩ int. 𝑎𝑓𝑑̂ ∩ int. 𝑑𝑒𝑐̂ ∩ int. 𝑏𝑓𝑐̂ ⟹ 𝑃 ∈ |𝛼𝐷 ∩ |𝛾𝐶 ∩ 𝛽𝐵 and
𝑃 ∈ |𝛿𝐴 ∩ |𝛾𝐶 ∩ |𝛽𝐵 ⟹ 𝑃 ∈ |𝛼𝐷 and 𝑃 ∈ |𝛽𝐵 and 𝑃 ∈ |𝛾𝐶 and 𝑃 ∈ |𝛿𝐶 ⟹ 𝑃 ∈
|𝛼𝐷 ∩ |𝛾𝐶 ∩ 𝛽𝐵 ∩ 𝛿𝐴 ⟹ 𝑃 ∈ int. [𝐴𝐵𝐶𝐷]. So 𝑀 ∈ [𝐴𝐵𝐶𝐷].
2. Following the inverse reasoning we show that [𝐴𝐵𝐶𝐷] ⊂ 𝑀 from where the
equality.

136
such that
such that 𝑁 ∈ |𝐶𝑃|, (𝐴𝐷𝐵) ∩ (𝐷𝑃𝐶) = 𝐷𝑃.
From 𝑁 ∈ |𝐶𝑃| and ∈ |𝐷𝑁|
𝑙𝑒𝑚𝑚𝑎
⇒ int. 𝐷𝑃𝐶 ⟹ 𝑀 ∈ int. 𝐷𝑃𝐶 ⟹ (∃)𝑄 ∈ |𝐷𝐶|.
So we showed that (∃)𝑃 ∈ |𝐴𝐵| and 𝑄 ∈ |𝐷𝐶| such that 𝑀 ∈ |𝑃𝑄|.
Let ℳ be the union of the open segments |𝑃𝑄|. We must prove that: int. [𝐴𝐵𝐶𝐷] =
ℳ through double inclusion.
1. Let 𝑀 ∈ int. [𝐴𝐵𝐶𝐷] ⟹ (∀)𝑃 ∈ |𝐴𝐵 and 𝑄 ∈ |𝐶𝐷 such that 𝑀 ∈ |𝑃𝑄| ⟹ 𝑀 ∈ ℳ so
int. [𝐴𝐵𝐶𝐷] ⊂ ℳ.
2. Let 𝑀 ∈ ℳ ⟹ (∃)𝑃 ∈ |𝐴𝐵| and 𝑄 ∈ |𝐶𝐷| such that 𝑀 ∈ |𝑃𝑄|. Points 𝐷, 𝐶 and 𝑃
determine plane (𝑃𝐷𝐶) and (𝑃𝐷𝐶) ∩ (𝐴𝐶𝐵) = 𝑃𝐶, (𝑃𝑈𝐶) ∩ (𝐴𝐷𝐵) = 𝑃𝐷.
As (∀)𝑄 ∈ |𝐶𝐷| such that 𝑀 ∈ |𝑃𝑄| ⟹ 𝑀 ∈ [𝑃𝐶𝐷] ⟹ |(∀)𝑅 ∈ |𝑃𝐶| such that 𝑀 ∈ |𝐷𝑅|.
If 𝑃 ∈ |𝐴𝐵| and 𝑅 ∈ |𝑃𝐶| ⟹ 𝑅 ∈ int. 𝐴𝐶𝐵 such that 𝑀 ∈ |𝐷𝑅| ⟹ 𝑀 int. [𝐴𝐵𝐶𝐷] ⟹
ℳ ⊂ int. [𝐴𝐵𝐶𝐷].
Working with closed segments we obtain that (∀)𝑅 ∈ [𝐴𝐶𝐵] such that 𝑀 ∈ [𝐷𝑅],
thus obtaining tetrahedron [𝐴𝐵𝐶𝐷].
Let 𝑀 ∈ [𝐴𝐵𝐶𝐷] ⟹ (∃) 𝑃 ∈ [𝐴𝐵𝐶] such that 𝑀 ∈ [𝐷𝑃].
Let 𝑁 ∈ [𝐴𝐵𝐶𝐷] ⟹ (∃) 𝑄 ∈ [𝐴𝐵𝐶] such that 𝑁 ∈ [𝐷𝑄].
The concurrent lines 𝐷𝑀 and 𝐷𝑁 determine angle 𝐷𝑀𝑁.
The surface of triangle 𝐷𝑃𝑄 is a convex set.

137
Let
such that 𝑂 ∈ [𝐷𝑅]. But [𝑃𝑄] ⊂ [𝐴𝐵𝐶] because 𝑃 ∈ [𝐴𝐵𝐶] ∩ 𝑄 ∈ [𝐴𝐵𝐶] and the
surface of the triangle is convex. So (∃)𝑅 ∈ [𝐴𝐵𝐶] such that
and the tetrahedron is a convex set.
Note: The tetrahedron can be regarded as the intersection of four closed half-
spaces which are convex sets.
Let ℳ be the union of the segments [𝑃𝑄] with 𝑃 ∈ ℳ1 and 𝑄 ∈ ℳ2.
Let 𝑥, 𝑥’ ∈ ℳ ⟹ (∀)𝑃 ∈ ℳ1 and 𝑄 ∈ ℳ2 such that 𝑥 ∈ [𝑃𝑄];
(∃)𝑃′ ∈ ℳ1 and 𝑄′ ∈ ℳ2 such that 𝑥′ ∈ [𝑃′𝑄′].
From 𝑃, 𝑃′ ∈ ℳ1 ⟹ [𝑃𝑃′]′
∈ ℳ1 which is a convex set.
From 𝑄, 𝑄′ ∈ ℳ2 ⟹ [𝑄𝑄′] ∈ ℳ2 which is a convex set.
The union of all the segments [𝑀𝑁] with 𝑀 ∈ [𝑃𝑃′] and 𝑁 ∈ [𝑄𝑄′] is tetrahedron
[𝑃𝑃′𝑄𝑄′] ⊂ ℳ.
So from 𝑥, 𝑥′ ∈ ℳ ⟹ |𝑥𝑥’| ⊂ ℳ, so set ℳ is convex.
The interior of the tetrahedron coincides with the union of segments |𝑃𝑄|, 𝑃 ∈
|𝐴𝐵| and 𝑄 ∈ |𝐶𝐷|, that is int. [𝐴𝐵𝐶𝐷] = {|𝑃𝑄| ∖ 𝑃 ∈ |𝐴𝐵|, 𝑄 ∈ |𝐶𝐷|}.
Let’s show that:

138
1. Let
2. Let
If we assume 𝑁 ∈ |𝐷𝑀| ⟹ |𝐷𝑀| ∩ (𝐴𝐵𝐶) ≠ ∅ ⟹ 𝑀 and 𝐷 are in different half-
spaces in relation to (𝐴𝐵𝐶) ⟹ 𝑀 ∉ (𝐴𝐵𝐶), 𝐷, false (it contradicts the hypothesis).
So
and the second inclusion is proved.
As regarding the tetrahedron: [𝐴𝐵𝐶𝐷] = {[𝑃𝑄] ∖ 𝑃 ∈ [𝐴𝐵] and 𝑄 ∈ [𝐶𝐷]}.

139
If
𝑃 = 𝐴, 𝑄 ∈ [𝐶𝐷], [𝑃𝑄] describes face [𝐴𝐷𝐶]
𝑃 = 𝐵, 𝑄 ∈ [𝐶𝐷], [𝑃𝑄] describes face [𝐵𝐷𝐶]
𝑄 = 𝐶, 𝑃 ∈ [𝐴𝐵], [𝑃𝑄] describes face [𝐴𝐵𝐶].
Because the triangular surfaces are convex sets and along with their two points 𝑃,
𝑄, segment [𝑃𝑄] is included in the respective surface.
So, if we add these two situations to the equality from the previous case, we
obtain:

140
Lines and Planes
159. Let 𝑑, 𝑑′
be two parallel lines. If the line 𝑑 is parallel to a plane 𝛼, show
that 𝑑′
||𝛼 or 𝑑′
⊂ 𝛼.
160. Consider a line 𝑑, parallel to the planes 𝛼 and 𝛽, which intersects after the
line 𝑎. Show that 𝑑‖𝑎.
161. Through a given line 𝑑, draw a parallel plane with another given line 𝑑′
.
Discuss the number of solutions.
162. Determine the union of the lines intersecting a given line 𝑑 and parallel
to another given line 𝑑′ (𝑑 ∦ 𝑑′).
163. Construct a line that meets two given lines and that is parallel to a third
given line. Discuss.
164. If a plane 𝛼 intersects the secant planes after parallel lines, then 𝛼 is
parallel to line 𝛽 ∩ 𝛾.
165. A variable plane cuts two parallel lines in points 𝑀 and 𝑁. Find the
geometrical locus of the middle of segment [𝑀𝑁].

141
166. Two lines are given. Through a given point, draw a parallel plane with
both lines. Discuss.
167. Construct a line passing through a given point, which is parallel to a
given plane and intersects a given line. Discuss.
168. Show that if triangles 𝐴𝐵𝐶 and 𝐴′𝐵′𝐶′, located in different planes, have
𝐴𝐵 ∥ 𝐴′𝐵′, 𝐴𝐶 ∥ 𝐴′𝐶′ and 𝐵𝐶 ∥ 𝐵′𝐶′, then lines 𝐴𝐴′, 𝐵𝐵′, 𝐶𝐶′ are concurrent or
parallel.
169. Show that, if two planes are parallel, then a plane intersecting one of
them after a line cuts the other one too.
170. Through the parallel lines 𝑑 and 𝑑′
we draw the planes 𝛼 and 𝛼′
distinct
from (𝑑, 𝑑′
). Show that 𝛼 ∥ 𝛼′
or (𝛼 ∩ 𝛼′
) ∥ 𝑑.
171. Given a plane 𝛼, a point 𝐴 ∈ 𝛼 and a line 𝑑 ⊂ 𝛼.
a. Construct a line 𝑑′
such that 𝑑′
⊂ 𝛼, 𝐴 ∈ 𝑑′
and 𝑑′
∥ 𝑑.
b. Construct a line through 𝐴 included in 𝛼, which forms with 𝑑 an angle
of a given measure 𝑎. How many solutions are there?
172. Show that relation 𝛼 ∥ 𝛽 defined on the set of planes is an equivalence
relation. Define the equivalence classes.

142
173. Consider on the set of all lines and planes the relation “𝑥 ∥ 𝑦” or 𝑥 = 𝑦,
where 𝑥 and 𝑦 are lines or planes. Have we defined an equivalence
relation?
174. Show that two parallel segments between parallel planes are concurrent.
175. Show that through two lines that are not contained in the same plane, we
can draw parallel planes in a unique way. Study also the situation when the
two lines are coplanar.
176. Let 𝛼 and 𝛽 be two parallel planes, 𝐴, 𝐵 ∈ 𝛼, and 𝐶𝐷 is a parallel line with
𝛼 and 𝛽. Lines 𝐶𝐴, 𝐶𝐵, 𝐷𝐵, 𝐷𝐴 cut plane 𝛽 respectively in 𝑀, 𝑁, 𝑃, 𝑄. Show
that these points are the vertices of a parallelogram.
177. Find the locus of the midpoints of the segments that have their
extremities in two parallel planes.

143
Solutions
Let
𝐴 ∈ 𝛼
𝑑 ∥ 𝛼
𝑑′′
∥ 𝑑
} ⟹ 𝑑′′
∥ 𝛼
𝑑′
∥ 𝑑
𝑑′′
∥ 𝑑
} ⟹ 𝑑′
∥ 𝑑′′
}
⟹ 𝑑′′
∥ 𝛼 or 𝑑′
⊂ 𝛼.
Let 𝐴 ∈ 𝑎 ⇒ 𝐴 ∈ 𝛼 ∩ 𝐴 ∈ 𝛽. We draw through A, 𝑑′
∥ 𝑑.
𝐴 ∈ 𝛼,
𝑑 ∥ 𝛼
𝑑′
∥ 𝑑
} ⟹ 𝑑′
⊂ 𝛼
𝐴 ∈ 𝛽,
𝑑 ∥ 𝛽
𝑑′
∥ 𝑑
} ⟹ 𝑑′
⊂ 𝛽
} ⟹
𝑑′
⊂ 𝛼 ⊂ 𝛽
𝛼 ∩ 𝛽 = 𝑎
} ⟹
𝑑′
= 𝑎
𝑑′
∥ 𝑑
} ⟹ 𝑎 ∥ 𝑑
a. If 𝑑 ∦ 𝑑′ there is only one solution and it can be obtained as it follows:
Let 𝐴 ∈ 𝑑. In the plane (𝐴, 𝑑′′
) we draw 𝑑′′
∥ 𝑑′
. The concurrent lines 𝑑 and 𝑑′′
determine plane 𝑎. As 𝑑′′
∥ 𝑑 ⟹ 𝑑 ∥ 𝛼, in the case of the non-coplanar lines.

144
b. If 𝑑 ∥ 𝑑′ || d'', (∃) infinite solutions. Any plane passing through 𝑑 is parallel to
𝑑′′
, with the exception of plane (𝑑, 𝑑′).
c. 𝑑 ∦ 𝑑′, but they are coplanar (∄) solutions.
Let 𝐴 ∈ 𝑑, we draw through 𝐴, 𝑑1 ∥ 𝑑′. We write 𝛼 = (𝑑, 𝑑1). As 𝑑1 ∥ 𝑑′ ⟹ 𝑑′ ∥ 𝛼.
Let 𝑀 ∈ 𝑑, arbitrary ⟹ 𝑀 ∈ 𝛼.
We draw
𝛿 ∥ 𝛿′
, 𝑀 ∈ 𝛿
𝑑′ ∥ 𝑎
} ⟹ 𝛿 ⊂ 𝛼, so all the parallel lines to 𝑑′ intersecting 𝑑 are
contained in plane 𝛼.
Let 𝛾 ⊂ 𝛼, 𝛾 ∥ 𝑑′
⟹ 𝛾 ∩ 𝑑 = 𝐵, so (∀) parallel to 𝑑′
from 𝛼 intersects 𝑑. Thus, the
plane 𝛼 represents the required union.

145
We draw 𝑑 through 𝑀 such that
𝑑 ∥ 𝑑′
𝑑′ ∥ 𝑑3
} ⟹ 𝑑 ∥ 𝑑3. According to previous
problem: 𝑑 ∩ 𝑑1 = {𝑁}. Therefore,
a. If 𝑑3 ∥ 𝑑1, the plane 𝛼 is unique, and if 𝑑2 ∩ 𝛼 ≠ ∅, the solution is unique.
b. If 𝑑1 ∥ 𝑑3, (∄)
𝑑 ∥ 𝑑1
𝑑 ∩ 𝑑1 ≠ ∅
, because it would mean that we can draw through a
point two parallel lines 𝑑, 𝑑1 to the same line 𝑑3. So there is no solution.
c. If 𝑑1 ∦ 𝑑3 and 𝑑2 ∩ 𝛼 ≠ ∅, all the parallel lines to 𝑑2 cutting 𝑑1 are on the plane 𝛼
and none of them can intersect 𝑑2, so the problem has no solution.
d. If 𝑑2 ⊂ 𝛼, 𝑑1 ∩ 𝑑2 ≠ ∅, let 𝑑1 ∩ 𝑑2 = {𝑂}, and the required line is parallel to 𝑑3
drawn through 𝑂 ⟹ one solution.
e. If 𝑑2 ⊂ 𝛼, 𝑑1 ∥ 𝑑2. The problem has infinite solutions, (∀) || to 𝑑3 which cuts 𝑑1,
also cuts 𝑑2.

146
The problem is reduced to the geometrical locus of the midpoints of the
segments that have extremities on two parallel lines. 𝑃 is such a point |𝑀𝑃| = |𝑃𝑁|.
We draw 𝐴𝐵 ⊥ 𝑑1 ⇒ 𝐴𝐵?𝑀𝑃𝐴̂ = 𝐵𝑃𝑁 ⟹̂ 𝛥𝑀𝐴𝑃 = 𝛥𝑁𝐵𝑃 ⟹ |𝑃𝐴| ≡ |𝑃𝐵| ⟹ ||𝐴𝑃||? ⟹
the geometrical locus is the parallel to 𝑑1 and 𝑑2 drawn on the mid-distance
between them. It can also be proved vice-versa.

147
Let 𝑑1 ∦ 𝑑2. In plane (𝑑, 𝑀) we draw 𝑑1
′
∥ 𝑑1, 𝑀 ∈ 𝑑1
′
. In plane (𝑑2 𝑀) we draw
𝑑2
′
∥ 𝑑2, 𝑀 ∈ 𝑑2
′
. We note 𝛼 = 𝑑1
′
𝑑2
′
the plane determined by two concurrent lines.
𝑀 ∈ 𝛼 the only solution.
Let 𝑑1 ∥ 𝑑2, 𝑁 ∉ 𝑑1, 𝑀 ∉ 𝑑2.
𝑑1 = 𝑑1
′
= 𝑑2
′
In this case 𝑑1
′
= 𝑑2
′
= 𝑑 and infinite planes pass through 𝑑;
𝑑2 ∥ 𝑑
𝑑1 ∥ 𝑑
} ⟹ 𝑑1, 𝑑2 are parallel lines with (∀) of the planes passing through 𝑑.
The problem has infinite solutions. But 𝑀 ∈ 𝑑1 or 𝑀 ∈ 𝑑2, the problem has no
solution because the plane can’t pass through a point of a line and be parallel to
that line.
Let 𝐴 be the given point, 𝛼 the given plane and 𝑑 the given line.
a. We assume that 𝑑 ∦ 𝛼, 𝑑 ∩ 𝛼 = {𝑀}. Let plane (𝑑𝐴) which has a common point
𝑀 with 𝛼 ⇒ (𝑑𝐴) ∩ 𝛼 = 𝑑′.
We draw in plane (𝑑𝐴) through point 𝐴 a parallel line to 𝑑′.

148
⟹ 𝑎 is the required line.
b. 𝑑 ∥ 𝛼, (𝑑𝐴) ∩ 𝛼 ≠ ∅.
Let (dA) ∩ α = α’
𝑑 ∥ 𝛼
} ⇒ 𝑑′ ∥ 𝑑
All the lines passing through 𝐴 and intersecting 𝑑 are contained in plane (𝑑𝐴). But
all these lines also cut 𝑑′ ∥ 𝑑, so they can’t be parallel to 𝛼. There is no solution.
c. 𝑑 ∥ 𝛼, (𝑑𝐴) ∩ 𝛼 = ∅.
Let 𝑀 ∈ 𝑑 and line 𝐴𝑀 ⊂ (𝑑𝐴); (𝑑𝐴) ∩ 𝛼 = ∅ ⟹ 𝐴𝑀 ∩ 𝛼 = ∅ ⟹ 𝐴𝑀 ∥ 𝛼, (∀)𝑀 ∈ 𝑑.
The problem has infinite solutions.
(𝐴𝐵𝐶) and (𝐴′𝐵′𝐶′) are distinct planes, thus the six points 𝐴, 𝐵, 𝐶, 𝐴′, 𝐵′, 𝐶′ can’t be
coplanar.
𝐴𝐵 ∥ 𝐴′𝐵’ ⟹ 𝐴, 𝐵, 𝐴′, 𝐵′ are coplanar.

149
The points are coplanar four by four, that is (𝐴𝐵𝐵′𝐴), (𝐴𝐶𝐶′𝐴′), (𝐵𝐶𝐶′𝐵′), and
determine four distinct planes. If we assumed that the planes coincide two by two,
it would result other 6 coplanar points and this is false.
In plane 𝐴𝐵𝐵′𝐴′, lines 𝐴𝐴′, 𝐵𝐵′ can be parallel or concurrent.
First we assume that:
is a common point to the 3 distinct planes, but the intersection of 3 distinct
planes can be only a point, a line or ∅. It can’t be a line because lines
⟹ are distinct if we assumed that two of them coincide, the 6 points would be
coplanar, thus there is no common line to all the three planes. There is one
possibility left, that is they have a common point 𝑆 and from
We assume 𝑑 ∩ 𝛽 = ∅ ⟹ 𝑑 ∥ 𝛽 ⟹ 𝑑 ∈ plane ∥ 𝛽 drawn through 𝐴 ⟹ 𝑑 ⊂ 𝛼, false.
So 𝑑 ∩ 𝛽 = {𝐵}.
Hypothesis: 𝛼 ∥ 𝛽, 𝛾 ∩ 𝛼 = 𝑑1.
Conclusion: 𝛾 ∩ 𝛽 = 𝑑2.
We assume that 𝛾 ∩ 𝛽 = ∅ ⟹ 𝛾 ∥ 𝛽.
Let

150
because from a point we can draw only one parallel plane with the given plane.
But this result is false, it contradicts the hypothesis 𝛾 ∩ 𝛼 = 𝑑1 so 𝛾 ∩ 𝛽 = 𝑑2.
Hypothesis: 𝑑 ∥ 𝑑′; 𝑑 ⊂ 𝛼; 𝑑′ ⊂ 𝛼′; 𝛼, 𝑎′ ≠ (𝑑𝑑′
).
Conclusion: 𝛼 ∥ 𝛼′ or 𝑑′′ ∥ 𝑑.
As 𝛼, 𝑎′ ≠ (𝑑𝑑′
) ⟹ 𝛼 ≠ 𝑎′.
If 𝑎 ∩ 𝑎′
= ∅ ⟹ 𝑎 ∥ 𝑎′.
If 𝑎 ∩ 𝑎′
= ∅ ⟹ 𝑎 ∩ 𝑎′
= 𝑑′′
.
If
𝑑 ∥ 𝑑′′
𝑑′ ⊂ 𝑎′
⟹
𝑑 ∥ 𝑎′
𝑑 ⊂ 𝑎
} ⟹ 𝑑′′‖𝑑.
a. If 𝐴 ∈ 𝑑, then 𝑑′ = 𝑑. If 𝐴 ∉ 𝑑, we draw through 𝐴, 𝑑′ ∥ 𝑑.

151
b. We draw 𝑑1 ⊂ 𝛼, 𝐴 ∈ 𝑑, such that 𝑚(𝑑1 𝑑′̂) = 𝑎 and 𝑑 ⊂ 𝛼, 𝐴 ∈ 𝑑2, such that
𝑚(𝑑2 𝑑′̂) = 𝑎, a line in each half-plane determined by 𝑑′
. So (∃) 2 solutions
excepting the situation 𝑎 = 0 or 𝑎 = 90 when (∃) only one solution.
𝛼 ∥ 𝛽 or 𝛼 = 𝛽 ⟺ 𝛼~𝛽
1. 𝛼 = 𝛼 ⟹ 𝛼~𝛼, the relation is reflexive;
2. 𝛼~𝛽 ⟹ 𝛽~ ∝, the relation is symmetric.
𝛼 ∥ 𝛽 or 𝛼 = 𝛽 ⟹ 𝛽 ∥ 𝛼 or 𝛽 = 𝛼 ⟹ 𝛽~ ∝;
3. 𝛼~𝛽 ∩ 𝛽~𝛾 ⟹ 𝛼~𝛾.
If 𝛼 = 𝛽 ∩ 𝛽~𝛾 ⟹ 𝛼~𝛾.
If
𝛼 ≠ 𝛽 and 𝛼~𝛽 ⟹ 𝛼 ∥ 𝛽
𝛽~𝛾 ⟹ 𝛽 = 𝛾 or 𝛽 ∥ 𝛾
} ⟹ 𝛼 ∥ 𝛾 ⟹ 𝛼~𝛾.
The equivalence class determined by plane 𝛼 is constructed of planes 𝛼′ with 𝛼′~𝛼,
that is of 𝛼 and all the parallel planes with 𝛼.
No, it is an equivalence relation, because the transitive property is not true. For
example, 𝑥 is a line, 𝑦 a plane, 𝑧 a line. From 𝑥 || 𝑦 and || 𝑧 ⇏ 𝑥 || 𝑧, lines 𝑥 and 𝑧
could be coplanar and concurrent or non-coplanar.

152
⟹ 𝐴𝐵𝐶𝐷 parallelogram.
So ||𝐴𝐶|| = ||𝐵𝐷||.
We consider 𝐴 ∈ 𝑑 and draw through it 𝑑1 ∥ 𝑑′
. We consider 𝐵 ∈ 𝑑′
and draw
𝑑2 ∥ 𝑑. Plane (𝑑1 𝑑2) ∥ (𝑑𝑑1), because two concurrent lines from the first plane are
parallel with two concurrent lines from the second plane.
When 𝑑 and 𝑑′ are coplanar, the four lines 𝑑, 𝑑1, 𝑑2 and 𝑑′ are coplanar and the
two planes coincide with the plane of the lines 𝑑 and 𝑑′.
Let planes:

153
From (1), (2), (3), (4) ⟹ 𝑀𝑁𝑃𝑄 parallelogram.
Let [𝐴𝐵] and [𝐶𝐷] be two segments, with 𝐴, 𝐶 ∈ 𝛼 and 𝐵, 𝐷 ∈ 𝛽 such that |𝐴𝑀| =
|𝑀𝐵| and |𝐶𝑁| = |𝑁𝐷|.
In plane (𝑀𝐶𝐷) we draw through 𝑀, 𝐸𝐹||𝐶𝐷 ⟹ 𝐸𝐶||𝐷𝐹 ⟹ 𝐸𝐹𝐷𝐶 parallelogram
⟹ |𝐸𝐹| ≡ |𝐶𝐷|.
Concurrent lines 𝐴𝐵 and 𝐸𝐹 determine a plane which cuts planes 𝛼 after 2 parallel
lines ⟹ 𝐸𝐴||𝐵𝐹.
In this plane, |𝐴𝑀| ≡ |𝐵𝑀|.
𝐸𝑀𝐴̂ ≡ 𝐵𝑀𝐹 (angles opposed at peak)̂
𝐸𝐴𝑀̂ ≡ 𝐹𝐵𝑀(alternate interior angles)̂
}
In parallelogram 𝐸𝐶𝐷𝐹,

154
So the segment connecting the midpoints of two of the segments with the
extremity in 𝛼 and 𝛽 is parallel to these planes. We also consider [𝐺𝐻] with 𝐺 ∈
𝛼, 𝐻 ∈ 𝛽 and |𝐺𝑄| ≡ |𝑄𝐻| and we show in the same way that 𝑂𝑀||𝛼 and 𝑂𝑀||𝛽. (2)
From (1) and (2) ⟹ 𝑀, 𝑁, 𝑄 are elements of a parallel plane to 𝛼 and 𝛽, marked by
𝛾.
Vice-versa, let’s show that any point from this plane is the midpoint of a segment,
with its extremities in 𝛼 and 𝛽.
Let segment [𝐴𝐵] with 𝐴 ∈ 𝛼 and 𝐵 ∈ 𝛽 and |𝐴𝑀| = |𝐵𝑀|. Through 𝑀, we draw the
parallel plane with 𝛼 and 𝛽 and in this plane we consider an arbitrary point 𝑂 ∈ 𝛾.
Through 𝑂 we draw a line such that 𝑑 ∩ 𝛼 = {𝐼} and 𝑑 ∩ 𝛽 = {𝐼}.
In plane (𝑂𝐴𝐵) we draw 𝐴′𝐵′ || 𝐴𝐵. Plane (𝐴𝐴′𝐵′𝐵) cuts the three parallel planes
after parallel lines ⟹
In plane (𝐴′𝐵′𝐼) ⟹ |𝐴′𝑂| ≡ |𝑂𝐵′| ⟹ 𝐼𝐴′ || 𝐵′𝐼 and thus
|𝐴′𝑂| ≡ |𝑂𝐵′|
𝐼𝑂𝐴′̂ ≡ 𝐼𝑂𝐵′̂
and 𝐼𝐴′𝑂̂ ≡
𝐼𝐵′𝑂̂ ⟹
𝑂 is the midpoint of a segment with extremities in planes 𝛼 and 𝛽.
Thus the geometrical locus is plane 𝛾, parallel to 𝛼 and 𝛽 and passing through the
mid-distance between 𝛼 and 𝛽.

155
Projections
178. Show that if lines 𝑑 and 𝑑′
are parallel, then pr 𝛼 𝑑 ∥ pr 𝛼 𝑑′
or pr 𝛼 𝑑 = pr 𝛼 𝑑′
.
What can we say about the projective planes of 𝑑 and 𝑑′
?
179. Show that the projection of a parallelogram on a plane is a parallelogram
or a segment.
180. Knowing that side [𝑂𝐴 of the right angle 𝐴𝑂𝐵 is parallel to a plane 𝛼,
show that the projection of 𝐴𝑂𝐵̂ onto the plane 𝛼 is a right angle.
181. Let 𝐴′𝐵′𝐶′ be the projection of ∆𝐴𝐵𝐶 onto a plane 𝛼. Show that the
centroid of ∆𝐴𝐵𝐶 is projected onto the centroid of ∆𝐴’𝐵’𝐶’. Is an analogous
result true for the orthocenter?
182. Given the non-coplanar points 𝐴, 𝐵, 𝐶, 𝐷, determine a plane on which the
points 𝐴, 𝐵, 𝐶, 𝐷 are projected onto the peaks of parallelogram.
183. Consider all triangles in space that are projected onto a plane 𝛼 after the
same triangle. Find the locus of the centroid.
184. Let 𝐴 be a point that is not on line 𝑑. Determine a plane 𝛼 such that pr 𝛼 𝑑
passes through pr 𝛼 𝐴.

156
185. Determine a plane onto which three given lines to be projected after
concurrent lines.
186. Let 𝛼, 𝛽 be planes that cut each other after a line 𝑎 and let 𝑑 be a
perpendicular line to 𝑎. Show that the projections of line 𝑑 onto 𝛼, 𝛽 are
concurrent.
187. Consider lines 𝑂𝐴, 𝑂𝐵, 𝑂𝐶 ⊥ two by two. We know that ||𝑂𝐴|| = 𝑎,
||𝑂𝐵|| = 𝑏, ||𝑂𝐶|| = 𝑐. Find the measure of the angle of planes (𝐴𝐵𝐶) and
(𝑂𝐴𝐵).
188. A line cuts two perpendicular planes 𝛼 and 𝛽 at 𝐴 and 𝐵. Let 𝐴′
and 𝐵′
be
the projections of points 𝐴 and 𝐵 onto line 𝛼 ∩ 𝛽.
a. Show that ||𝐴𝐵||² = ||𝐴𝐴′||² + ||𝐴′𝐵′||² + ||𝐵′𝐵||²;
b. If 𝑎, 𝑏, 𝑐 are the measures of the angles of line 𝐴𝐵 with planes 𝛼, 𝛽 and
with 𝛼 ∩ 𝛽, then cos 𝑐
‖𝐴′ 𝐵′‖
‖𝐴𝐵‖
and sin2
𝑎 + sin2
𝑏 = sin2
𝑐.
189. Let 𝐴𝐵𝐶 be a triangle located in a plane 𝛼, 𝐴′𝐵′𝐶′ the projection of
∆𝐴′𝐵′𝐶′ onto plane 𝛼. We mark with 𝑆, 𝑆′
, 𝑆′′
the areas of ∆𝐴𝐵𝐶,
∆𝐴′
𝐵′
𝐶′
, ∆𝐴ʺ𝐵ʺ𝐶ʺ, show that 𝑆′ is proportional mean between 𝑆 and 𝑆′′
.
190. A trihedral [𝐴𝐵𝐶𝐷] has |𝐴𝐶| ≡ |𝐴𝐷| ≡ |𝐵𝐶| ≡ |𝐵𝐷|. 𝑀, 𝑁 are the midpoints
of edges [𝐴𝐵], [𝐶𝐷], show that:
a. 𝑀𝑁 ⊥ 𝐴𝐵, 𝑀𝑁 ⊥ 𝐶𝐷, 𝐴𝐵 ⊥ 𝐶𝐷

157
b. If 𝐴′, 𝐵′, 𝐶′, 𝐷′ are the feet of the perpendicular lines drawn to the peaks
𝐴, 𝐵, 𝐶, 𝐷 on the opposite faces of the tetrahedron, points 𝐵, 𝐴′, 𝑁 are
collinear and so are 𝐴, 𝐵′, 𝑁; 𝐷, 𝐶′, 𝑀; 𝐶, 𝐷′, 𝑀.
c. 𝐴𝐴′, 𝐵𝐵′, 𝑀𝑁 and 𝐶𝐶′, 𝐷𝐷′, 𝑀𝑁 are groups of three concurrent lines.
191. If rays [𝑂𝐴 and [𝑂𝐵 with their origin in plane 𝛼, 𝑂𝐴 ⊥ 𝛼, then the two rays
form an acute or an obtuse angle, depending if they are or are not on the
same side of plane 𝛼.
192. Show that the 6 mediator planes of the edges of a tetrahedron have a
common point. Through this point pass the perpendicular lines to the faces
of the tetrahedron, drawn through the centers of the circles of these faces.
193. Let 𝑑 and 𝑑′ be two non-coplanar lines. Show that (∃) unique points 𝐴 ∈
𝑑, 𝐴′ ∈ 𝑑′ such that 𝐴𝐴′ ⊥ 𝑑 and 𝐴𝐴′ ⊥ 𝑑′. The line 𝐴𝐴′ is called the common
perpendicular of lines 𝑑 and 𝑑′.
194. Consider the notations from the previous problem. Let 𝑀 ∈ 𝑑, 𝑀′ ∈ 𝑑′.
Show that ‖𝐴𝐴′‖ ≤ ‖𝑀𝑀′‖. The equality is possible only if 𝑀 = 𝐴, 𝑀′ = 𝐴′.
195. Let 𝐴𝐴′ be the common ⊥ of non-coplanar lines 𝑑, 𝑑′′
and 𝑀 ∈ 𝑑, 𝑀′ ∈ 𝑑′
such that |𝐴𝑀| ≡ |𝐴′𝑀′|. Find the locus of the midpoint of segment [𝑀𝑀′].
196. Consider a tetrahedron 𝑉𝐴𝐵𝐶 with the following properties. 𝐴𝐵𝐶 is an
equilateral triangle of side 𝑎, (𝐴𝐵𝐶) ⊥ (𝑉𝐵𝐶), the planes (𝑉𝐴𝐶) and (𝑉𝐴𝐵)

158
form with plane (𝐴𝐵𝐶) angles of 60°. Find the distance from point 𝑉 to
plane (𝐴𝐵𝐶).
197. All the edges of a trihedral are of length a. Show that a peak is projected
onto the opposite face in its centroid. Find the measure of the dihedral
angles determined by two faces.
198. Let 𝐷𝐸 be a perpendicular line to the plane of the square 𝐴𝐵𝐶𝐷. Knowing
that ‖𝐵𝐸‖ = 𝑙 and that the measure of the angle formed by [𝐵𝐸 and (𝐴𝐵𝐶)
is 𝛽, determine the length of segment 𝐴𝐸 and the angle of [𝐴𝐸 with plane
(𝐴𝐵𝐶).
199. Line 𝐶𝐷 ⊥ plane of the equilateral ∆𝐴𝐵𝐶 of side 𝑎, and [𝐴𝐷 and [𝐵𝐷 form
with plane (𝐴𝐵𝐶) angles of measure 𝛽. Find the angle of planes (𝐴𝐵𝐶) and
𝐴𝐵𝐷.
200. Given plane 𝛼 and ∆𝐴𝐵𝐶, ∆𝐴’𝐵’𝐶’ that are not on this plane. Determine a
∆𝐷𝐸𝐹, located on 𝛼 such that on one side lines 𝐴𝐷, 𝐵𝐸, 𝐶𝐹 and on the
other side lines 𝐴′𝐷, 𝐵′𝐸, 𝐶′𝐹 are concurrent.

159
Solutions
Let 𝑑 ∥ 𝑑′
, 𝛽 the projective plane of 𝑑.
We assume that 𝑑′
⊄ 𝛽, which means that is plane 𝑑, 𝑑′
⊥ 𝛼, ⟹ the projective
plane of 𝑑′
is 𝛽′
. We want to show that pr 𝑎 𝑑 ∥ pr 𝑎 𝑑′
. We assume that pr 𝑎 𝑑 ∩
pr 𝑎 𝑑′
= {𝑃} ⇒ (∃) 𝑀 ∈ 𝑑 such that pr 𝑎 𝑀 = 𝑃 and (∃)𝑀′
∈ 𝑑′
such that pr 𝑎 𝑀′
= 𝑃.
⟹
𝑃𝑀 ⊥ 𝛼
𝑃𝑀′
⊥ 𝛼
} ⟹ in the point 𝑃 on plane 𝛼 we can draw two distinct perpendicular
lines. False.
If 𝛽 is the projective plane of 𝑑 and 𝛽 of 𝑑′
, then 𝛽 ∥ 𝛽′
, because if they had a
common point their projections should be elements of pr 𝑎 𝑑 and pr 𝑎 𝑑′
, and thus
they wouldn’t be anymore parallel lines.
If 𝑑′
⊂ 𝛽 or 𝑑 ⊂ 𝛽′
, that is (𝑑, 𝑑′) ⊥ 𝛼 ⟹ 𝑑 and 𝑑′
have the same projective plane
⟹ pr 𝑎 𝑑 = pr 𝑎 𝑑′
.

160
We assume that 𝐴𝐵𝐶𝐷 ⊥ 𝛼. Let 𝐴′, 𝐵′, 𝐶′, 𝐷′ be the projections of points 𝐴, 𝐵, 𝐶, 𝐷.
𝐴𝐵 ∥ 𝐷𝐶
𝑝𝑟1
⇒ 𝐴′
𝐵′
∥ 𝐷′
𝐶′
𝐴𝐷 ∥ 𝐵𝐶 ⟹ 𝐴′
𝐷′
∥ 𝐵′
𝐷′
} ⟹ 𝐴′, 𝐵′, 𝐶′, 𝐷′ parallelogram.
If (𝐴𝐵𝐶𝐷) ⊥ 𝛼 ⇒ the projection 𝐴′, 𝐵′, 𝐶′, 𝐷′
∈ the line (𝐴𝐵𝐶𝐷) ∩ 𝛼 ⟹ the
projection of the parallelogram is a segment.
If 𝑂𝐴||𝛼 ⟹ proj 𝛼 𝑂𝐴||𝑂𝐴 ⟹ 𝑂′𝐴′||𝑂𝐴 because (∀) a plane which passes through
𝑂𝐴 cuts the plane 𝛼 after a parallel to 𝑂𝐴.
⟹ 𝑂′𝐴′ ⊥ 𝑂′𝐵′ ⟹ 𝐴′𝑂′𝐵′̂ is a right angle.
In the trapezoid 𝐵𝐶𝐶′𝐵′ (𝐵𝐵′||𝐶𝐶′),

161
⟹ 𝐴′𝑀′ is a median.
𝑀𝑀′ ∥ 𝐴𝐴′ ⟹ 𝑀𝑀′𝐴′𝐴 trapezoid
‖𝐴𝐺‖
‖𝐺𝑀‖
= 2, 𝐺𝐺′ ∥ 𝐴𝐴′
}
‖𝐴′𝐺‖
‖𝐺′𝑀′‖
=
‖𝐴𝐺‖
‖𝐺𝑀‖
= 2
⟹ 𝐺′ is on median A'M' at 2/3 from the peak and 1/3 from the base.
Generally no, because the right angle 𝐴𝑀𝐶 should be projected after a
right angle. The same thing is true for another height. This is achieved if the
sides of the ∆ are parallel to the plane.
Let 𝐴, 𝐵, 𝐶, 𝐷 be the 4 non-coplanar points and 𝑀, 𝑁 midpoints of
segments |𝐴𝐵| and |𝐶𝐷|.
𝑀 and 𝑁 determine a line and let a plane 𝛼 ⊥ 𝑀𝑁, 𝑀 and 𝑁 are projected
in the same point 𝑂 onto 𝛼.
⟹ 𝐴′𝐵′𝐶′𝐷′ a parallelogram.

162
Let 𝐴′𝐵′𝐶′, 𝐴′′𝐵′′𝐶′′ two triangles of this type, with the following property:
⟹ 𝐺′
𝐺 = 𝐺𝐺′′
⟹ 𝐺′′
, 𝐺′
, 𝐺 are collinear.
Due to the fact that by projection the ratio is maintained, we show that 𝐺′′ is the
centroid of 𝐴, 𝐵, 𝐶.
Let 𝑀 ∈ 𝑑 and 𝐴 ∉ 𝑑. The two points determine a line and let 𝛼 be a
perpendicular plane to this line, 𝐴𝑀 ⊥ 𝛼 ⟹ 𝐴 and 𝑀 are projected onto 𝛼 in the
same point 𝐴′ through which also passes projα 𝑑 = projα 𝐴 ∈ projα 𝑑.

163
We determine a line which meets the three lines in the following way.
Let
Let now a plane
Let 𝛼 ∩ 𝛽 = 𝑎 and 𝑀 ∈ 𝑑. We project this point onto 𝛼 and 𝛽:
⟹ 𝑎 ⊥ onto the projective plane of 𝑑 onto 𝛽.

164
Let
⟹ 𝑂𝑀′
∩ 𝑂𝑀′′
= {𝑂}, so the two projections are concurrent.
the angle of planes (𝐴𝐵𝐶) and (𝑂𝐴𝐵) is 𝑂𝑀𝐶̂ = 𝛼.
Let 𝛼 ∩ 𝛽 = a and 𝐴𝐴′ ⊥ 𝑎, 𝐵𝐵′ ⊥ 𝑎.

165
As
⟹ ∢ of line 𝐴𝐵 with 𝛼, 𝐵𝐴𝐵′̂ = 𝑎
⟹ ∢ of line 𝐴𝐵 with 𝛽, 𝐴𝐵𝐴′ = 𝑏
In the plane 𝛽 we draw through 𝐵 a parallel line to 𝑎 and through 𝐴′
a parallel line
to 𝐵𝐵′
. Their intersection is 𝐶, and ||𝐴′
𝐵′|| = ||𝐵𝐶||, ||𝐵𝐵′|| = ||𝐴′
𝐶||. The angle of line
𝐴𝐵 with 𝛼 is 𝐴𝐵𝐶̂ = 𝑐.
As 𝐴𝐴’ ⊥ 𝛽 ⟹ 𝐴𝐴′
⊥ 𝐴′
𝐶 ⟹ ‖𝐴𝐶‖2
= ‖𝐴𝐴′‖2
+ ‖𝐴′𝐶‖2
= ‖𝐴𝐴′‖2
+ ‖𝐵′𝐵‖2
(1)
𝐵′
𝐵𝐶𝐴 rectangle
⟹ ∆𝐴𝐶𝐵 is right in 𝐶.
We divide the relation (1) with ||𝐴𝐵||²:

166
a.
|𝐴𝐶| ≡ |𝐵𝐶| ⟹ ∆𝐴𝐶𝐵 isosceles
𝐶𝑀 median
} ⟹ 𝐶𝑀 ⊥ 𝐴𝐵 (1)
|𝐴𝐷| ≡ |𝐵𝐶| ⟹ ∆𝐴𝐵𝐷 isosceles
𝐷𝑀 median
} ⟹ 𝐷𝑀 ⊥ 𝐴𝐵 (2)
From (1) and (2) ⟹ 𝐴𝐵 ⊥ (𝐷𝑀𝐶) =
𝐴𝐵 ⊥ 𝑀𝑁
𝐴𝐵 ⊥ 𝐷𝐶
}
|𝐵𝐶| ≡ |𝐵𝐷|
𝐵𝑁 median
} ⟹ 𝐵𝑁 ⊥ 𝐷𝐶
|𝐴𝐷| ≡ |𝐴𝐶|
𝐴𝑁 median
} ⟹ 𝐴𝑁 ⊥ 𝐷𝐶
} ⟹ 𝐷𝐶 ⊥ (𝐴𝐵𝑁) ⟹ 𝐷𝐶 ⊥ 𝑀𝑁
b.
⟹ 𝐴′
∈ 𝐵𝑁 ⟹ 𝐵, 𝐴′
, 𝑁 are collinear.
In the same way:
From (𝐴𝐷𝐶) ⊥ (𝐴𝐵𝑁) ⟹ 𝐴, 𝐵′, 𝑁 collinear
(𝐴𝐵𝐶) ⊥ (𝐷𝑀𝐶) ⟹ 𝑀, 𝐷′, 𝐶 collinear
(𝐴𝐵𝐷) ⊥ (𝐷𝑀𝐶) ⟹ 𝐷, 𝐶′, 𝑀 collinear
C. At point a. we’ve shown that 𝑀𝑁 ⊥ 𝐴𝐵
𝐴𝐴′, 𝐵𝐵′ and 𝑀𝑁 are heights in ∆𝐴𝐵𝑁, so they are concurrent lines.
In the same way, 𝐷𝐷′, 𝐶𝐶′, 𝑀𝑁 will be heights in ∆𝐷𝑀𝐶.

167
We assume that [𝑂𝐴, [𝑂𝐵 are on the same side of plane 𝛼.
We draw
(∃) plane (𝐴𝐷, 𝐵𝐵′) = 𝛽 ⟹ |𝑂𝐴, |𝑂𝐵 are in the same half-plane.
In plane 𝛽 we have 𝑚(𝐴𝑂𝐵̂) = 900
= 𝑚(𝐵𝑂𝐵̂′) < 900
⟹ 𝐴𝑂𝐵̂ acute.
We assume that [𝑂𝐴 and [𝑂𝐵 are in different half-planes in relation to 𝛼 ⟹ 𝐴 and 𝐵 are
in different half-planes in relation to 𝑂𝐵′ in plane 𝛽 ⟹ |𝑂𝐵’ ⊂ int. 𝐵𝑂𝐴̂ ⟹ 𝑚(𝐴𝑂𝐵̂) =
900
+ 𝑚(𝐵𝑂𝐵’̂ ) > 900
⟹ 𝐴𝑂𝐵 obtuse.
We know the locus of the points in space equally distant from the peaks of ∆𝐵𝐶𝐷
is the perpendicular line 𝑑 to the pl. ∆ in the center of the circumscribed circle of
this ∆, marked with 𝑂. We draw the mediator plane of side |𝐴𝐶|, which intersects
this ⊥ 𝑑 at point 𝑂. Then, point 𝑂 is equally distant from all the peaks of the
tetrahedron ||𝑂𝐴|| = ||𝑂𝐵|| = ||𝑂𝐶|| = ||𝑂𝐷||. We connect 𝑂 with midpoint 𝐸 of side
|𝐴𝐵|. From |𝑂𝐴| ≡ |𝑂𝐵| ⟹ ∆𝑂𝐴𝐵 isosceles ⟹ 𝑂𝐶 ⊥ 𝐴𝐵 (1).

168
We project 𝑂 onto plane (𝐴𝐵𝐷) in point 𝑂2.
As
|𝑂𝐴| ≡ |𝑂𝐵| ≡ |𝑂𝐷|
|𝑂)𝑂2 common side
} ⟹ ∆𝑂𝐴𝑂2 = ∆𝑂𝐵𝑂2 = ∆𝑂𝐷𝑂2
⟹ |𝑂2 𝐴| ≡ |𝐵𝑂2| ≡ |𝐷𝑂2| ⟹
⟹ 𝑂2 is the center of the circumscribed circle of ∆𝐴𝐵𝐷. We show in the same
way that 𝑂 is also projected on the other faces onto the centers of the
circumscribed circles, thus through 𝑂 pass all the perpendicular lines to the faces of
the tetrahedron. These lines are drawn through the centers of the circumscribed
circles. So b. is proved.
From |𝑂2 𝐴| ≡ |𝑂𝐵2| ⟹ ∆𝑂2 𝐴𝐵 isosceles 𝑂2 𝐸 ⊥ 𝐴𝐵 (2)
From (1) and (2) ⟹
𝐴𝐵 ⊥ (𝐸𝑂2 𝑂)
|𝐴𝐸| ≡ |𝐸𝐵|
} ⟹
⟹ (𝐸𝑂2 𝑂) is a mediator plane of side |𝐴𝐵| and passes through 𝑂 and the
intersection of the 3 mediator planes of sides |𝐵𝐶|, |𝐶𝐷|, |𝐵𝐷| belongs to line 𝑑, thus
O is the common point for the 6 mediator planes of the edges of a tetrahedron.
Let 𝑀 ∈ 𝑑 and 𝛿||𝑑′, 𝑀 ∈ 𝛿. Let = (𝑑, 𝛿) ⟹ 𝑑′||𝛼.
Let
= {𝐴} otherwise 𝑑 and 𝑑′ would be parallel, thus coplanar. Let 𝛽 be the projective
plane of line
In plane 𝛽 we construct a perpendicular to 𝑑′′ in point 𝐴 and

169
We draw
We can obtain the equality only when 𝑀 = 𝐴 and 𝑀′ = 𝐴′.
Let 𝑀 ∈ 𝑑, 𝑀′ ∈ 𝑑′ such that |𝐴𝑀| ≡ |𝐴′𝑀′|. Let 𝑑′′
= pr 𝛼 𝑑′ and 𝑀′𝑀′′ ⊥ 𝑑′′ ⟹
𝑀′𝑀′′ ⊥ 𝛼 ⟹ 𝑀′𝑀′′ ⊥ 𝑀′′𝑀.
⟹ ∆𝐴𝑀𝑀′ isosceles.
Let 𝑃 be the midpoint of |𝑀𝑀′| and 𝑃′ = pr 𝛼 𝑃 ⟹ 𝑃𝑃′ ∥ 𝑀′𝑀′′ ⟹ 𝑃′ is the midpoint
of 𝑀𝑀′′, ∆𝐴𝑀𝑀′′ isosceles ⟹ [𝐴𝑃′ the bisector of 𝑀′𝐴𝑀̂ . (𝑃𝑃′) is midline in
∆𝑀′
𝑀ʺ𝑀 ⟹ ‖𝑃𝑃′‖ =
1
2
‖𝑀′
𝑀′′
=
1
2
‖𝐴′𝐴‖ = constant.
Thus, the point is at a constant distance from line 𝐴𝑃′, thus on a parallel line to
this line, located in the ⊥ plane 𝛼, which passes through 𝐴𝑃′.
When 𝑀 = 𝐴 and 𝑀′ = 𝐴′ ⟹ ||𝐴𝑀|| = ||𝑁′𝐴′ = 0 ⟹ 𝑃 = 𝑅, where 𝑅 is the midpoint
of segment |𝐴𝐴′|. So the locus passes through 𝑅 and because
⟹ 𝑅𝑃 is contained in the mediator plane of segment |𝐴𝐴′|.
So 𝑅𝑃 is the intersection of the mediator plane of segment |𝐴𝐴′| with the ⊥ plane
to 𝛼, passing through one of the bisectors of the angles determined by 𝑑 and 𝑑′,
we obtain one more line contained by the mediator plane of [𝐴𝐴′], the parallel line
with the other bisector of the angles determined by 𝑑 and 𝑑′′.
So the locus will be formed by two perpendicular lines.

170
Vice-versa, let 𝑄 ∈ 𝑅𝑃 a (∀) point on this line and 𝑄’ = pr 𝛼 𝑄 ⟹ 𝑄′ ∈ |𝐴𝑃′ bisector.
We draw 𝑁𝑁′′ ⊥ 𝐴𝑄′ and because 𝐴𝑄′ is both bisector and height ⟹ ∆𝐴𝑁𝑁′′
isosceles ⟹ |𝐴𝑄′| median ⟹ |𝑁𝑄′| ≡ |𝑄′𝑁′′|.
We draw
As
⟹ |𝑄′𝑄| midline in ∆𝑁𝑁′
𝑁′′
⟹ 𝑄, 𝑁′
, 𝑁 collinear and |𝑄𝑁′| ≡ |𝑄𝑁|.
where α = (ABC).
VD common side
coplanar

171
Let
⟹ 𝑂 is the center of the circumscribed circle and as ∆𝐴𝐵𝐶 is equilateral ⟹ 𝑂 is the
centroid ⟹
In ∆𝐴𝐸𝐵, right in 𝐴:
||VO|| common

172
𝐶𝐸 ⊥ 𝐵𝐴
𝐷𝐸 ⊥ 𝐴𝐵
} ⟹ ∢pl. (𝐴𝐵𝐶) and 𝐴𝐵𝐷 are 𝑚(𝐷𝐸𝐶̂).
𝐴𝐵𝐶 equilateral ⟹ ‖𝐶𝐸‖ =
𝑎√3
2
.
We consider the problem solved and we take on plane 𝛼, ∆𝐷𝐸𝐹, then points 𝑂
and 𝑂′ which are not located on 𝛼.
We also construct lines |𝐷𝑂, |𝐹𝑂, |𝐸𝑂 respectively |𝐷𝑂′, |𝐹𝑂′, |𝐸𝑂′. On these rays we
take ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. Obviously, the way we have constructed the lines 𝐴𝐷, 𝐵𝐸, 𝐶𝐹
shows that they intersect at 𝑂. We extend lines 𝐵𝐴, 𝐵𝐶, 𝐶𝐴 until they intersect plane

173
𝛼 at points 𝐵, 𝐶 respectively 𝐴. Then, we extend lines 𝐶′𝐴′, 𝐶′𝐵′, 𝐴′𝐵′ until they
intersect plane 𝛼 at points 𝐴2, 𝐶2 respectively 𝐵2.
Obviously, points 𝐴1, 𝐵1, 𝐶1 are collinear (because ∈ 𝛼 ∩ (𝐴𝐵𝐶)) and 𝐴2, 𝐵2, 𝐶2 are as
well collinear (because ∈ 𝛼 ∩ (𝐴′𝐵′𝐶′)).
On the other side, points 𝐷, 𝐹, 𝐴1, 𝐴2 are collinear because:
thus collinear (1)
⟹ 𝐷, 𝐹, 𝐴2 collinear (2)
From (1) and (2) ⟹ 𝐷, 𝐹, 𝐴1, 𝐴2 collinear. Similarly 𝐶, 𝐸, 𝐹, 𝐶2 collinear and
𝐵1, 𝐸, 𝐷, 𝐷2 collinear.
Consequently, 𝐷𝐸𝐹 is at the intersection of lines 𝐴1 𝐴2, 𝐶1 𝐶2 , 𝐵1 𝐵2 on plane 𝛼,
thus uniquely determined.

174
Review Problems
201. Find the position of the third peak of the equilateral triangle, the affixes
of two peaks being 𝑧1 = 1, 𝑧2 = 2 + 𝑖.
202. Let 𝑧1, 𝑧2, 𝑧3 be three complex numbers, not equal to 0, + two by 2, and
of equal moduli. Prove that if 𝑧1 + 𝑧2 𝑧3, 𝑧2 + 𝑧3 𝑧1, 𝑧2 + 𝑧 1 𝑧3 ∈ 𝑅 ⇒ 𝑧1 𝑧2 𝑧3 = 1.
203. We mark by 𝐺 the set of 𝑛 roots of the unit, 𝐺 = {𝜀0, 𝜀1, … , 𝜀 𝑛−1}. Prove
that:
a. 𝜀𝑖 ∙ 𝜀𝑗 ∈ 𝐺, (∀) 𝑖, 𝑗 ∈ {0, 1, … , 𝑛 − 1};
b. 𝜀𝑖
−1
∈ 𝐺, (∀ ) 𝑖 ∈ {0, 1, … , 𝑛 − 1}.
204. Let the equation 𝑎𝑧² + 𝑏𝑧² + 𝑐 = 0, 𝑎, 𝑏, 𝑐 ∈ 𝐶 and arg𝑎 + arg𝑐 = 2arg𝑏, and
|𝑎| + |𝑐| = |𝑏|. Show that the given equation has at list one root of unity.
205. Let 𝑧1, 𝑧2, 𝑧3 be three complex numbers, not equal to 0, such that |𝑧1| =
|𝑧2| = |𝑧3|.
a. Prove that (∃) complex numbers 𝛼 and 𝛽 such that 𝑧2 = 𝛼𝑧1, 𝑧3 = 𝛽𝑧2
and |𝛼| = |𝛽| = 1;
b. Solve the equation 𝛼² + 𝛽²– 𝛼 ∙ 𝛽– 𝛼– 𝛽 + 1 = 0 in relation to one of the
unknowns.
c. Possibly using the results from 𝑎. and 𝑏., prove that if 𝑧1
2
+ 𝑧2
2
+ 𝑧3
2
=
𝑧1 𝑧2 + 𝑧2 𝑧3 + 𝑧1 𝑧3 , then we have 𝑧1 = 𝑧2 = 𝑧3 or the numbers 𝑧1, 𝑧2, 𝑧3
are affixes of the peaks of an equilateral ∆.

175
206. Draw a plane through two given lines, such that their line of intersection
to be contained in a given plane.
207. Let 𝑎, 𝑏, 𝑐 be three lines with a common point and 𝑃 a point not located
on any of them. Show that planes (𝑃𝑎), (𝑃𝑏), (𝑃𝑐) contain a common line.
208. Let 𝐴, 𝐵, 𝐶, 𝐷 be points and 𝛼 a plane separating points 𝐴 and 𝐵, 𝐴 and 𝐶,
𝐶 and 𝐷. Show that 𝛼 ∩ |𝐵𝐷| ≠ ∅ and 𝛼 ∩ |𝐴𝐷| = ∅.
209. On edges 𝑎, 𝑏, 𝑐 of a trihedral angle with its peak 𝑂, take points 𝐴, 𝐵, 𝐶; let
then 𝐷 ∈ |𝐵𝐶| and 𝐸 ∈ |𝐴𝐷|. Show that |𝑂𝐸 ⊂ 𝑖𝑛𝑡. ∠𝑎𝑏𝑐.
210. Show that the following sets are convex: the interior of a trihedral angle,
a tetrahedron without an edge (without a face).
211. Let 𝐴, 𝐵, 𝐶, 𝐷 be four non-coplanar points and 𝐸, 𝐹, 𝐺, 𝐻 the midpoints of
segments [𝐴𝐵], [𝐵𝐶], [𝐶𝐷], [𝐷𝐴]. Show that 𝐸𝐹 || (𝐴𝐶𝐷) and points 𝐸, 𝐹, 𝐺, 𝐻
are coplanar.
212. On lines 𝑑, 𝑑′ consider distinct points 𝐴, 𝐵, 𝐶; 𝐴′, 𝐵′, 𝐶′ respectively. Show
that we can draw through lines 𝐴𝐴′, 𝐵𝐵′, 𝐶𝐶′ three parallel planes if and only
if
‖𝐴𝐵‖
‖𝐴′ 𝐵′‖
=
‖𝐵𝐶‖
‖𝐵′ 𝐶′‖
.

176
213. Let 𝑀, 𝑀′ be each mobile points on the non-coplanar lines 𝑑, 𝑑′. Find the
locus of points 𝑃 that divide segment |𝑀𝑀′| in a given ratio.
214. Construct a line that meets three given lines, respectively in 𝑀, 𝑁, 𝑃 and
for which
‖𝑀𝑁‖
‖𝑁𝑃‖
to be given ratio.
215. Find the locus of the peak 𝑃 of the triangle 𝑀, 𝑁, 𝑃 if its sides remain
parallel to three fixed lines, the peak 𝑀 describes a given line 𝑑, and the
peak 𝑁 ∈ a given plane 𝛼.
216. On the edges [𝑂𝐴, [𝑂𝐵, [𝑂𝐶 of a trihedral angle we consider points 𝑀, 𝑁, 𝑃
such that ‖𝑂𝑀‖ = 𝜆‖𝑂𝐴‖, ‖𝑂𝑁‖ = 𝜆‖𝑂𝐵‖, ‖𝑂𝑃‖ = 𝜆‖𝑂𝐶‖, where 𝜆 is a
positive variable number. Show the locus of the centroid of triangle 𝑀𝑁𝑃.
217. 𝐴𝐵𝐶𝐷 and 𝐴1 𝐵1 𝐶1 𝐷1 are two parallelograms in space. We take the points
𝐴2, 𝐵2, 𝐶2, 𝐷2 which divide segments [𝐴𝐴1], [𝐵𝐵1], [𝐶𝐶1], [𝐷𝐷1] in the same
ratio. Show that 𝐴2 𝐵2 𝐶2 𝐷2 is a parallelogram.
218. The lines 𝑑, 𝑑′ are given, which cut a given plane 𝛼 in 𝐴 and 𝐴′
. Construct
the points 𝑀, 𝑀′ on 𝑑, 𝑑′ such that 𝑀𝑀′ ∥ 𝛼 and segment [𝑀𝑀′] to have a
given length 𝑙. Discuss.
219. Construct a line which passes through a given point 𝐴 and that is
perpendicular to two given lines 𝑑 and 𝑑′.

177
220. Show that there exist three lines with a common point, perpendicular two
by two.
221. Let 𝑎 𝑏, 𝑐, 𝑑 four lines with a common point, 𝑑 is perpendicular to 𝑎 𝑏, 𝑐.
Show that lines 𝑎, 𝑏, 𝑐 are coplanar.
222. Show that there do not exist four lines with a common point that are
perpendicular two by two.
223. Let 𝑑 ⊥ 𝛼 and 𝑑′
∥ 𝑑. Show that 𝑑′
⊥ 𝛼.
224. Show that two distinct perpendicular lines on a plane are parallel.
225. Let 𝑑 ⊥ 𝛼 and 𝑑′[∥ 𝛼. Show that 𝑑′
⊥ 𝑑.
226. Show that two perpendicular planes on the same line are parallel with
each other.
227. Show that the locus of the points equally distant from two distinct points
𝐴 and 𝐵 is a perpendicular plane to 𝐴𝐵, passing through midpoint 𝑂 of the
segment [𝐴𝐵] (called mediator plane of [𝐴𝐵]).

178
228. Find the locus of the points in space equally distant from the peaks of a
triangle 𝐴𝐵𝐶.
229. The plane 𝛼 and the points 𝐴 ∈ 𝛼, 𝐵 ∉ 𝛼 are given. A variable line 𝑑 passes
through 𝐴 and it is contained in plane 𝛼. Find the locus of the ⊥ feet from
𝐵 to 𝑑.
230. A line 𝛼, and a point 𝐴 ∉ 𝛼 are given. Find the locus of the feet of the
perpendicular lines from 𝐴 to planes passing through 𝛼.
231. Consider a plane 𝛼 that passes through the midpoint of segment [𝐴𝐵].
Show that points 𝐴 and 𝐵 are equally distant from plane 𝛼.
232. Through a given point, draw a line that intersects a given line and is ⊥ to
another given line.
233. Let 𝛼 and 𝛽 be two distinct planes and the line 𝑑 their intersection. Let 𝑀
be a point that is not located on 𝛼 ∪ 𝛽. We draw the lines 𝑀𝑀1 and 𝑀𝑀2 ⊥
on 𝛼 and 𝛽. Show that the line 𝑑 is ⊥ to (𝑀𝑀1 𝑀𝑀2).
234. A plane 𝛼 and a point 𝐴, 𝐴 ∉ 𝛼 are given. Find the locus of points 𝑀 ∈ 𝛼
such that segment |𝐴𝑀| has a given length.

179
235. Let 𝑂, 𝐴, 𝐵, 𝐶 be four points such that 𝑂𝐴 ⊥ 𝑂𝐵 ⊥ 𝑂𝐶 ⊥ 𝐷𝐴 and we write
𝑎 = ‖𝑂𝐴‖, 𝑏 = ‖𝑂𝐵‖, 𝑐 = ‖𝑂𝐶‖.
a. Find the length of the sides of ∆𝐴𝐵𝐶 in relation to 𝑎, 𝑏, 𝑐;
b. Find 𝜎[𝐴𝐵𝐶] and demonstrate the relation 𝜎[𝐴𝐵𝐶]2
= 𝜎[𝐷𝐴𝐵]2
+
𝜎[𝑂𝐵𝐶]2
+ 𝜎[𝑂𝐶𝐴]2
;
c. Show that the orthogonal projection of point 𝑂 on plane (𝐴𝐵𝐶) is the
orthocenter 𝐻 of ∆𝐴𝐵𝐶;
d. Find the distance ‖𝑂𝐻‖.
236. Consider non-coplanar points 𝐴, 𝐵, 𝐶, 𝐷 and lines 𝐴𝐴′, 𝐵𝐵′, 𝐶𝐶′, 𝐷𝐷′
perpendicular to (𝐵𝐶𝐷), (𝐴𝐶𝐷), (𝐴𝐵𝐷). Show that if lines 𝐴𝐴′ and 𝐵𝐵′ are
concurrent, then lines 𝐶𝐶′, 𝐷𝐷′ are coplanar.
237. Let 𝐴, 𝐵, 𝐶, 𝐷 four non-coplanar points. Show that 𝐴𝐵 ⊥ 𝐶𝐷 and 𝐴𝐶 ⊥ 𝐵𝐷
⟹ 𝐴𝐷 ⊥ 𝐵𝐶.
238. On the edges of a triangle with its peak 𝑂, take the points 𝐴, 𝐵, 𝐶 such
that |𝑂𝐴| ≡ |𝑂𝐵| ≡ |𝑂𝐶|. Show that the ⊥ foot in 𝑂 to the plane (𝐴𝐵𝐶)
coincides with the point of intersection of the bisectors ∆𝐴𝐵𝐶.
239. Let a peak 𝐴 of the isosceles triangle 𝐴𝐵𝐶 (|𝐴𝐵| ≡ |𝐴𝐶|) be the
orthogonal projection onto 𝐴′ on a plane 𝛼 which passes through 𝐵𝐶. Show
that 𝐵𝐴′ 𝐶̂ > 𝐵𝐴𝐶̂.
240. With the notes of Theorem 1, let [𝐴𝐵′
be the opposite ray to [𝐴𝐵′′
. Show
that for any point 𝑀 ∈ 𝛼– [𝐴𝐵′′
we have 𝐵′′ 𝐴𝐵̂ > 𝑀𝐴𝐵̂ .

180
241. Let 𝛼 be a plane, 𝐴 ∈ 𝛼 and 𝐵 and 𝐶 two points on the same side of 𝛼
such that 𝐴𝐶 ⊥ 𝛼. Show that 𝐶𝐴𝐵̂ is the complement of the angle formed
by [𝐴𝐵 with 𝛼.
242. Let 𝛼′𝛽′ be a trihedral angle with edge 𝑚 and 𝐴 ∈ 𝑚. Show that of all the
rays with origin at 𝐴 and contained in half-plane 𝛽′, the one that forms
with plane 𝛼 the biggest possible angle is that ⊥ 𝑝 ∈ 𝑚 (its support is
called the line with the largest slope of 𝛽 in relation to 𝛼).
243. Let 𝛼 be a plane, 𝜎 a closed half-plane, bordered by 𝛼, 𝛼′
a half-plane
contained in 𝛼 and 𝑎 a real number between 00
and 1800
. Show that there
is only one half-space 𝛽′
that has common border with 𝛼′
such that 𝛽′
⊂ 𝜎
and 𝑚(𝛼′
𝛽′
) = 𝑎.
244. Let (𝛼′ 𝛽′̂) be a proper dihedral angle. Construct a half-plane 𝛾′
such that
𝑚(𝛼′ 𝛽′̂) = 𝑚(𝛾′ 𝛽′̂). Show that the problem has two solutions, one of which
is located in the int. 𝛼′ 𝛽′̂ (called bisector half-plane of 𝛼′ 𝛽′̂).
245. Show that the locus of the points equally distant from two secant planes
𝛼, 𝛽 is formed by two ⊥ planes, namely by the union of the bisector planes
of the dihedral angles 𝛼 and 𝛽.
246. If 𝛼 and 𝛽 are two planes, 𝑄 ∈ 𝛽 and 𝑑 ⊥ through 𝑄 on 𝛼. Show that 𝑑 ⊂
𝛽.

181
247. Consider a line 𝑑 ⊂ 𝛼. Show that the union of the ⊥ lines to 𝛼, which
intersect line 𝑑, is a plane ⊥ 𝛼.
248. Find the locus of the points equally distant from two concurrent lines.
249. Show that a plane 𝛼 ⊥ to two secant planes is ⊥ to their intersection.
250. Let 𝐴 be a point that is not on plane 𝛼. Find the intersection of all the
planes that contain point 𝐴 and are ⊥ to plane 𝛼.
251. From a given point draw a ⊥ plane to two given planes.
252. Intersect a dihedral angle with a plane as the angle of sections is right.
253. Show that a line 𝑑 and a plane 𝛼, which are perpendicular to another
plane, are parallel or line 𝑑 is contained in 𝛼.
254. If three planes are ⊥ to a plane, they intersect two by two after lines
𝑎, 𝑏, 𝑐. Show that 𝑎 ∥ 𝑏 ∥ 𝑐.
255. From a point 𝐴 we draw perpendicular lines 𝐴𝐵 and 𝐴𝐶 to the planes of
the faces of a dihedral angle 𝛼′ 𝛽′̂. Show that 𝑚(𝐵𝐴𝐶̂) = 𝑚(𝛼′ 𝛽′̂) or
𝑚(𝐵𝐴𝐶̂) = 1800
− 𝑚(𝛼′ 𝛽′̂).

182
Solutions
𝑀1 − 𝑧1 = 1
𝑀2 − 𝑧1 = 2 + 𝑖
𝑀1 − 𝑧1 = 𝑥 + 𝑦𝑖
∆𝑀1 𝑀2 𝑀3 equilateral ⟹ ‖𝑀1 𝑀2‖ = ‖𝑀1 𝑀3‖ = ‖𝑀2 𝑀3‖ ⇒ |𝑧2 − 𝑧1| = |𝑧3 − 𝑧2| =
|𝑧1 − 𝑧3|
⇒ √2 = √(𝑥 − 2)2 + (𝑦 − 1)2 ⇒ {
(𝑥 − 2)2
+ (𝑦 − 1)2
= 2
(1 − 𝑥)2
+ 𝑦2
= 2
⇒ {
𝑥 + 𝑦 = 2
𝑥2
+ 𝑦2
− 2𝑥 = 1
⇒ 𝑦 = 2 − 𝑥
𝑥2
+ 4 + 𝑥2
− 4𝑥 − 2𝑥 = 1 ⇒ 𝑥1,2 =
3 ± √3
2
⇒
[
𝑦1 =
1 − √3
2
𝑦2
1 + √3
2
Thus: 𝑀3 (
3+√3
2
,
1−√3
2
) or 𝑀3 (
3−√3
2
,
1+√3
2
).
There are two solutions!
𝑧1 = 𝑟(cos 𝑡1 + 𝑖 sin 𝑡1)
𝑧1 ≠ 𝑧2 ≠ 𝑧3 ⇒ 𝑡1 ≠ 𝑡2 ≠ 𝑡3
{
𝑧1 + 𝑧2 𝑧3 ∈ ℝ ⇒ sin 𝑡1 + 𝑟 sin(𝑡2 + 𝑡3) = 0
𝑧2 + 𝑧3 𝑧1 ∈ ℝ ⇒ sin 𝑡2 + 𝑟 sin(𝑡1 + 𝑡3) = 0
𝑧3 + 𝑧1 𝑧2 ∈ ℝ ⇒ sin 𝑡3 + 𝑟 sin(𝑡1 + 𝑡2) = 0
⟹
{
sin 𝑡1(1 − 𝑟 cos 𝑡) + 𝑟 sin 𝑡 ∙ cos 𝑡1 = 0
𝑡1 ≠ 𝑡2 ≠ 𝑡3
These equalities are simultaneously true only if 1 − 𝑟 ∙ cos 𝑡 = 0 and 𝑟 ∙
sin 𝑡 = 0, as 𝑟 ≠ 0 ⇒ sin 𝑡 = 0 ⇒ 𝑡 = 0 ⇒ cos 𝑡 = 1 ⇒ 1 − 𝑟 = 0 ⇒ 𝑟 = 1, so
𝑧1 𝑧2 𝑧3 = 1 ∙ (cos 0 + sin 0) = 1.

183
a. 𝜀 𝑘 =
2𝑘𝜋
𝑛
+ 𝑖 sin
2𝑘𝜋
𝑛
, 𝑘 ∈ {0, 1, … , 𝑛 − 1}.
So
𝜀𝑖 = cos
2𝑖𝜋
𝑛
+ 𝑖 sin
2𝑖𝜋
𝑛
𝜀𝑗 = cos
2𝑗𝜋
𝑛
+ 𝑖 sin
2𝑗𝜋
𝑛
} ⟹ 𝜀𝑖 𝜀𝑗 = cos
2𝜋(𝑖+𝑗)
𝑛
+ 𝑖 sin
2𝜋(𝑖+𝑗)
𝑛
, 𝑖, 𝑗 ∈ {0, 1, … , 𝑛 − 1}.
1) 𝑖 + 𝑗 < 𝑛 − 1 ⟹ 𝑖 + 𝑗 = 𝑘 ∈ {0, 1, … , 𝑛 − 1} ⟹ 𝜀𝑖 𝜀𝑗 = 𝜀 𝑘 ∈ 𝐺;
2) 𝑖 + 𝑗 = 𝑛 ⟹ 𝜀𝑖 𝜀𝑗 = cos 2𝜋 + 𝑖 sin 2𝜋 = 1 = 𝜀 𝑜 ∈ 𝐺;
3) 𝑖 + 𝑗 > 𝑛 ⟹ 𝑖 + 𝑗 = 𝑛 ∙ 𝑚 + 𝑟, 0 ≤ 𝑟 < 𝑛, 𝜀𝑖 𝜀𝑗 = cos
2𝜋(𝑛∙𝑚+𝑟)
𝑛
+ 𝑖 sin
2𝜋(𝑛∙𝑚+𝑟)
𝑛
=
cos(2𝜋𝑚 +
2𝜋𝑟
𝑛
) + 𝑖 sin (2𝜋𝑚 +
2𝜋𝑟
𝑛
) = cos
2𝜋𝑟
𝑛
+ 𝑖 sin
2𝜋𝑟
𝑛
= 𝜀 𝑟 ∈ 𝐺.
b. 𝜀𝑖 = cos
2𝜋𝑖
𝑛
+ 𝑖 sin
2𝜋𝑖
𝑛
𝜀𝑖
−1
=
1
𝜀 𝑖
=
cos 0+𝑖 sin0
cos
2𝜋𝑖
𝑛
+𝑖 sin
2𝜋𝑖
𝑛
= cos(−
2𝜋𝑖
𝑛
) + 𝑖 sin(−
2𝜋𝑖
𝑛
) = cos (2𝜋 −
2𝜋𝑖
𝑛
) +
𝑖 sin (2𝜋 −
2𝜋𝑖
𝑛
) = cos
2𝜋𝑛−2𝜋𝑖
𝑛
+ 𝑖 sin
2𝜋𝑛−2𝜋𝑖
𝑛
= cos
2𝜋(𝑛−1)
𝑛
+ 𝑖 sin
2𝜋(𝑛−1)
𝑛
,
𝑖 ∈ {0, 1, … , 𝑛 − 1}.
If 𝑖 = 0 ⟹ 𝑛 − 𝑖 = 𝑛 ⟹ 𝜀0
−1
= 𝜀0 ∈ 𝑔.
If 𝑖 ≠ 0 ⟹ 𝑛 − 𝑖 ≤ 𝑛 − 1 ⟹ ℎ = 𝑛 − 𝑖 ∈ {0, 1, … , 𝑛 − 1} ⟹ 𝜖−1
= cos
2𝜋ℎ
𝑛
+
𝑖 sin
2𝜋ℎ
𝑛
∈ 𝐺.
{
𝑎 = 𝑟1(cos 𝑡1 + 𝑖 sin 𝑡1)
𝑏 = 𝑟2(cos 𝑡2 + 𝑖 sin 𝑡2)
𝑐 = 𝑟3(cos 𝑡3 + 𝑖 sin 𝑡3)
arg𝑎 + arg𝑐 = 2arg𝑏 ⟹ 𝑡1 + 𝑡3 = 2𝑡2
and |𝑎| + |𝑐| = |𝑏| ⟹ 𝑟1 + 𝑟3 = 𝑟2
𝑎𝑧2
+ 𝑏𝑧 + 𝑐 = 0 ⟹ 𝑧1,2 =
−𝑏 ± √𝑏2 − 4𝑎𝑐
2𝑎
=
−𝑟2(cos 𝑡2 + 𝑖 sin 𝑡2) ± √𝑟2
2(cos2𝑡2 + 𝑖 sin2𝑡2) − 4𝑟1 𝑟3(cos(𝑡1 + 𝑡3) + 𝑖 sin(𝑡1 + 𝑡3))
2𝑟1(cos 𝑡1 + 𝑖 sin 𝑡1)
=
−𝑟2(cos 𝑡2 + 𝑖 sin 𝑡2) ± √(cos 2𝑡2 + 𝑖 sin 2𝑡2)(𝑟2
2
− 4𝑟1 𝑟3)
But 𝑟1 + 𝑟3 = 𝑟2 ⟹ 𝑟2
2
= 𝑟1
2
+ 𝑟1
2
+ 𝑟3
2
+ 2𝑟1 𝑟3 ⟹ 𝑟2
2
− 4𝑟1 𝑟3 = 𝑟1
2
+ 𝑟1
2
+ 𝑟3
2
+ 2𝑟1 𝑟3 −
4𝑟1 𝑟3 = (𝑟1 − 𝑟3)2
.

184
Therefore:
𝑧1,2 =
−𝑟2(cos 𝑡2 + 𝑖 sin 𝑡2) ± (cos 𝑡2 + 𝑖 sin 𝑡2)(𝑟1 − 𝑟3)
We observe that:
𝑧2 =
(cos 𝑡2+𝑖 sin 𝑡2)(−2𝑟1)
2𝑟1(cos 𝑡1+𝑖 sin 𝑡1)
= −[cos(𝑡2 − 𝑡1) + 𝑖 sin(𝑡2 − 𝑡1)] = cos[𝜋 + (𝑡2 − 𝑡1)] +
𝑖 sin[𝜋 + 𝑡2 − 𝑡1] and 𝑡2 = 1.
Let
Let
So 𝛼 is determined.
So 𝛽 is determined.
If we work with reduced arguments, then 𝑡4 = 𝑡2 − 𝑡1 or 𝑡4 = 𝑡2 − 𝑡1 + 2𝜋, in the
same way 𝑡5.

185
According to a. (∃) the complex numbers of modulus 1, 𝛼 and 𝛽 such that 𝑧2 = 𝛼𝑧1
and 𝑧3 = 𝛽𝑧1.
In the given relation, by substitution we obtain:
𝛼 = 1 and 𝛽 = 1 verify this equality, so in this case 𝑧2 = 𝑧3 = 𝑧1.
According to point b.,
where 𝛽 = 𝑥 + 𝑖𝑦, when
We construct the system:
The initial solution leads us to 𝑧1 = 𝑧2 = 𝑧3.
and gives
By substituting,

186
|𝛼| = 2 does not comply with the condition |𝛼| = 1.
But
so
If
then
and then
If
are on the circle with radius 𝑟 and the arguments are
they are the peaks of an equilateral triangle.
a. We assume that 𝑑 ∩ 𝛼 ≠ ∅ and 𝑑′ ∩ 𝛼 = {𝐵}.

187
Let 𝑑 ∩ 𝛼 = {𝐴} and 𝑑′ ∩ 𝛼 = {𝐵} and the planes determined by pairs of concurrent
lines (𝑑, 𝐴𝐵); (𝑑′, 𝐴𝐵).
We remark that these are the required planes, because
b. We assume 𝑑 ∩ 𝛼 = {𝐴} and 𝑑′||𝛼.
We draw through 𝐴, in plane 𝛼, line 𝑑′||𝑑 and we consider planes (𝑑, 𝑑′′) and
(𝑑′, 𝑑′′) and we remark that
c. We assume 𝑑 ∩ 𝛼 = ∅ and 𝑑′ ∩ 𝛼 = ∅ and 𝑑′ ∈ direction 𝑑.
Let 𝐴 ∈ 𝛼 and 𝑑′′||𝑑 ⟹ 𝑑′′||𝑑′ and the planes are (𝑑, 𝑑′′) and (𝑑′, 𝑑′′). The reasoning
is the same as above.

188
If 𝛼 separates points 𝐴 and 𝐵, it means they are in different half-spaces and let
𝜎 = |𝛼𝐴 and 𝜎′ = |𝛼𝐵.
Because 𝛼 separates 𝐴 and 𝐶 ⟹ 𝐶 ∈ 𝜎′.
Because 𝛼 separates 𝐶 and 𝐷 ⟹ 𝐷 ∈ 𝜎.
From 𝐵 ∈ 𝜎′ and 𝐷 ∈ 𝜎 ⟹ 𝛼 separates points 𝐵 and 𝐷
From 𝐴 ∈ 𝜎 and 𝐷 ∈ 𝜎 ⟹ |𝐵𝐷| ∩ 𝛼 = ∅.

189
From (1), (2), (3)
a. int. (|𝑉𝐴, |𝑉𝐵̂ , |𝑉𝐶) = |(𝑉𝐴𝐵), 𝐶 ∩ |(𝑉𝐵𝐶), 𝐴 ∩ |(𝑉𝐴𝐶), 𝐵 is thus an intersection of
convex set and thus the interior of a trihedron is a convex set.
b. Tetrahedron [𝑉𝐴𝐵𝐶] without edge [𝐴𝐶]. We mark with ℳ1 = [𝐴𝐵𝐶]– [𝐴𝐶] = [𝐴𝐵, 𝐶 ∩
[𝐵𝐶, 𝐴 ∩ |𝐴𝐶, 𝐵 is thus a convex set, being intersection of convex sets.
is a convex set.
In the same way
is a convex set, where
But [𝑉𝐴𝐵𝐶] – [𝐴𝐶] =
and thus it is a convex set as intersection of convex sets.
c. Tetrahedron [𝑉𝐴𝐵𝐶] without face [𝐴𝐵𝐶]
𝑉 is thus intersection of convex sets ⟹ is a convex set.

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In plane (𝐵𝐴𝐶) we have 𝐸𝐹||𝐴𝐶. In plane (𝐷𝐴𝐶) we have 𝐴𝐶 ⊂ (𝐷𝐴𝐶) ⟹ 𝐸𝐹||(𝐷𝐴𝐶).
In this plane we also have 𝐻𝐺||𝐴𝐶. So 𝐸𝐹||𝐻𝐺 ⟹ 𝐸, 𝐹, 𝐺, 𝐻 are coplanar and because
‖𝐸𝐹‖ =
‖𝐴𝐶‖
2
= ‖𝐻𝐺‖ ⟹ 𝐸𝐹𝐺𝐻 is a parallelogram.
We assume we have 𝛼||𝛽||𝛾 such that 𝐴𝐴’ ⊂ 𝛼, 𝐵𝐵’ ⊂ , 𝐶𝐶’ ⊂ 𝛾.
We draw through 𝐴′ a parallel line with 𝑑: 𝑑′′||𝑑. As 𝑑 intersects all the 3 planes
𝐴′ ⊂ 𝑑′′ at 𝐴, 𝐵, 𝐶 ⟹ and its || 𝑑’’ cuts them at 𝐴′, 𝐵′′, 𝐶′′.
Because
Let plane (𝑑′
, 𝑑′′ ). Because this plane has in common with planes 𝛼, 𝛽, 𝛾 the points
𝐴′, 𝐵′′, 𝐶′′ and because 𝛼 || 𝛽 || 𝛾 ⟹ it intersects them after the parallel lines
Taking into consideration (1) and (2)
The vice-versa can be similarly proved.

191
Let
such that
and
such that
So
according to problem 7, three planes can be drawn || 𝛽 || 𝛼 || 𝛾 such that
and 𝑃𝑃′ ⊂ 𝛼.
So by marking 𝑃 and letting 𝑃′ variable, 𝑃′ ∈ a parallel plane with the two lines,
which passes through 𝑃. It is known that this plane is unique, because by drawing
through 𝑃 parallel lines to 𝑑 and 𝑑’ in order to obtain this plane, it is well
determined by 2 concurrent lines.
Vice-versa: Let 𝑃 ∈ 𝛼, that is the plane passing through 𝑃 and it is parallel to 𝑑
and 𝑑’.

192
(𝑃′′, 𝑑) determines a plane, and (𝑃′′, 𝑑′) determines a plane ⟹ the two planes,
which have a common point, intersect after a line(𝑃′′, 𝑑)∩ (𝑃′′, 𝑑′) = 𝑄𝑄′ where 𝑄 ∈
𝑑 and 𝑄′ ∈ 𝑑′.
Because
such that 𝑀𝑄 ⊂ 𝛽, 𝛽||𝛼.
Because
such that 𝑀′𝑄′ ⊂ 𝛾, 𝛾||𝛼.
So the required locus is a parallel plane with 𝑑 and 𝑑’.
We consider the plane, which according to a previous problem, represents the
locus of the points dividing the segments with extremities on lines 𝑑1 and 𝑑3 in a
given ratio 𝑘. To obtain this plane, we take a point 𝐴 ∈ 𝑑1, 𝐵 ∈ 𝑑3 and point 𝐶 ∈ 𝐴𝐵
such that
‖𝐴𝐶‖
‖𝐶𝐵‖
= 𝑘. Through this point 𝐶 we draw two parallel lines 𝑑1 and 𝑑3 which
determine the above mentioned plane 𝛼.
Let 𝑑2 ∩ 𝛼 = {𝑁}. We must determine a segment that passes through 𝑁 and with
its extremities on 𝑑1 and 𝑑3, respectively at 𝑀 and 𝑃. As the required line passes
through 𝑁 and 𝑀
The same line must pass through 𝑁 and 𝑃 and because
𝑀, 𝑁, 𝑃 collinear.

193
From (1) and (2)
Then, according to previous problem 8:
and the required line is 𝑀𝑃.
Let ∆𝑀𝑁𝑃 such that
Let ∆𝑀′𝑁′𝑃′ such that
Line 𝑀𝑃 generates a plane 𝛽, being parallel to a fixed direction 𝑑1 and it is based
on a given line 𝑑. In the same way, the line 𝑀𝑁 generates a plane 𝛾, parallel to a
fixed direction 𝑑2, and based on a given line 𝑑. As 𝑑 is contained by 𝛾 ⟹ 𝑂 is a
common point for 𝛼 and 𝛾 ⟹ 𝛼 ∩ 𝛾 ≠ ∅ ⟹ 𝛼 ∩ 𝛾 = 𝑑′, 𝑂 ∈ 𝑑′.
(∀) the considered ∆, so 𝑁 also describes a line 𝑑′ ⊂ 𝛼.
Because plane 𝛾 is well determined by line 𝑑 and direction 𝑑2, is fixed, so 𝑑′ = 𝛼 ∩
𝛾 is fixed.
In the same way, 𝑃𝑁 will generate a plane 𝛿, moving parallel to the fixed direction
𝑑3 and being based on the given line 𝑑’.
As

194
(∀)𝑃 variable peak, 𝑃 ∈ 𝑑′′.
Thus, in the given conditions, for any ∆𝑀𝑁𝑃, peak 𝑃 ∈ 𝑑′′.
Vice-versa, let 𝑃′ ∈ 𝑑′′. On plane (𝑑′, 𝑑′′) we draw 𝑃′, 𝑀′||𝑃𝑀 ⟹ (𝑀′𝑃′𝑁′)||(𝑃𝑀𝑁) ⟹
(𝑑𝑑′) the intersection of two parallel planes after parallel lines 𝑀′𝑁′||𝑀𝑁 and the so
constructed ∆𝑀′𝑃′𝑁′ has its sides parallel to the three fixed lines, has 𝑀′ ∈ 𝑑 and
𝑁′ ∈ 𝛼, so it is one of the triangles given in the text.
So the locus is line 𝑑′′. We’ve seen how it can be constructed and it passes
through 𝑂.
In the situation when 𝐷||𝛼 we obtain
In this case the locus is a parallel line with 𝑑.
Let 𝑀𝑁𝑃 and 𝑀′𝑁′𝑃′ such that

195
We assume 𝛼 ∩ 𝛽 = 𝑑 and let 𝑑 ∩ (𝑀𝑁𝑃) = {𝑂} and 𝑑 ∩ (𝑀′𝑁′𝑃′) = {𝑂′} ⟹
a plane cuts the parallel planes after parallel lines.
In the same way, 𝑂𝑁||𝑂′𝑁′ and because
We use the property: Let 𝜋1 and 𝜋2 2 parallel planes and 𝐴, 𝐵, 𝐶 ⊂ 𝜋1 and 𝐴′𝐵′𝐶′ ⊂
𝜋2, 𝐴𝐵 ∥ 𝐴′𝐵′,
Let’s show that 𝐵𝐶||𝐵′𝐶′. Indeed (𝐵𝐵′𝐶′) is a plane which intersects the 2 planes
after parallel lines.
Applying in (1) this property ⟹ 𝑂𝑃||𝑂′𝑃′. Maintaining 𝑂𝑃 fixed and letting 𝑃’
variable, always 𝑂𝑃||𝑂′𝑃′.=, so 𝑂′𝑃′ generates a plane which passes through 𝑑. We
assume 𝛽||𝛼.
𝑀𝑁𝑁′𝑀′ parallelogram

196
Considering 𝑃′ fix and 𝑃 variable ⟹ 𝑃𝑃′||𝛼 and the set of parallel lines drawn
𝑃𝑃′||𝛽 to a plane through an exterior point is a parallel plane with the given plane.
So the locus is a parallel plane with 𝛼 and 𝛽.
In plane 𝐷𝐴𝐶 we have:
In plane 𝐷𝐴𝐵 we have:
In plane 𝑂𝐵𝐶 we have:
From 𝑃𝑀||𝐴𝐶 and 𝑃𝑁||𝐵𝐶 ⟹ (𝑀𝑁𝑅)||(𝐴𝐵𝐶).
Let 𝑄 and 𝐷 be midpoints of sides |𝑀𝑁| and |𝐴𝐵|.
are collinear.

197
Concurrent lines 𝑂𝐷 and 𝑂𝐶 determine a plane which cuts the parallel planes
are collinear.
So 𝐺’ ∈ |𝑂𝐺 ⟹ the required locus is ray |𝑂𝐺.
Vice-versa: we take a point on |𝑂𝐺, 𝐺′′, and draw through it a parallel plane
to (𝐴𝐵𝐶), plane (𝑀′′, 𝑁′′, 𝑃′′), similar triangles are formed and the ratios from
the hypothesis appear.
Let 𝐴2, 𝐵2, 𝐶2, 𝐷2 such that
Mark on lines 𝐴𝐷1 and 𝐵𝐶1 points 𝑀 and 𝑁 such that
From
Next is
The same,
As

198
we obtain
From
is a parallelogram.
is parallelogram.
So
is a parallelogram.

199
We draw through 𝐴′ a line 𝑑′′||𝑑. We draw two parallel planes with 𝛼, which will
intersect the three lines in 𝐵′, 𝐵∗
, 𝐵 and 𝐶′, 𝐶∗
, 𝐶. Plane (𝑑, 𝑑∗
) intersects planes 𝛼,
(𝐵′𝐵∗
𝐵), (𝐶′𝐶∗
𝐶) after parallel lines
= ‖𝐵𝐵∗‖ = ‖𝐶′𝐶∗‖.
Plane (𝑑′, 𝑑∗
) intersects parallel planes (𝐵′𝐵∗
𝐵), (𝐶′𝐶∗
𝐶) after parallel lines
So (∀) parallel plane with 𝛼 we construct, the newly obtained triangle has a side of
𝛼 length and the corresponding angle to 𝐵′𝐵∗ 𝐵̂ is constant. We mark with a line
that position of the plane, for which the opposite length of the required angle is 𝑙.
With the compass spike at 𝐶 and with a radius equal with 𝑙, we trace a circle arc
that cuts segment | 𝐶′𝐶∗
| at 𝑁 or line 𝐶′𝐶∗
. Through 𝑁 we draw at (𝑑′, 𝑑∗
) a parallel
line to 𝑑∗
which precisely meets 𝑑′ in a point 𝑀′. Through 𝑀′, we draw the || plane
to 𝛼, which will intersect the three lines in 𝑀, 𝑀′, 𝑀∗
.
is a parallelogram.
⟹ 𝐶𝑁𝑀′𝑀 is a parallelogram.
and line 𝑀𝑀′, located in a parallel plane to 𝛼, is parallel to 𝛼.
Discussion:
Assuming the plane (𝐶′𝐶∗
𝐶) is variable, as | 𝐶𝐶∗
| and 𝐶𝐶′𝐶∗̂ are constant, then
𝑑(𝐶′𝐶∗
𝐶) = 𝑏 = also constant
If 𝑙 < 𝑑 we don’t have any solution.
If 𝑙 = 𝑑 (∃) a solution, the circle of radius 𝑙, is tangent to 𝐶′𝐶∗
.
If 𝑙 > 𝑑 (∃ ) two solutions: circle of radius 𝑙, cuts 𝐶′𝐶∗
at two points 𝑁 and 𝑃.

200
We draw through 𝐴 planes 𝛼 ⊥ 𝑑 and 𝛼′ ⊥ 𝑑′.
As 𝐴 is a common point
⟹ 𝛼 ∩ 𝛼′
= ∆⟹ 𝐴 ∈ ∆.
⟹ ∆ the required line
If 𝛼 ≠ 𝛼′ - we have only one solution.
If 𝛼 = 𝛼′ (∀) line from 𝛼 which passes through 𝐴 corresponds to the problem, so
(∃) infinite solutions.
Let 𝑑1 ⊥ 𝑑2 two concurrent perpendicular lines, 𝑑1 ∩ 𝑑2 = {𝑂}. They determine a
plane 𝛼 = (𝑑1, 𝑑2) and 𝑂 ∈ 𝛼. We construct on 𝛼 in 𝑂.

201
We use the reductio ad absurdum method.
Let 𝑑 ⊥ 𝑎, 𝑑 ⊥ 𝑎, 𝑑 ⊥ 𝑐. We assume that these lines are not coplanar. Let 𝛼 =
(𝑏, 𝑐), 𝛼′ = (𝑎, 𝑏), 𝛼 ≠ 𝛼′. Then 𝑑 ⊥ 𝛼, 𝑑 ⊥ 𝛼’.
Thus through point 𝑂, 2 perpendicular planes to 𝑑 can be drawn. False ⟹ 𝑎, 𝑏, 𝑐 are
coplanar.
By reductio ad absurdum:
Let 𝑎 ∩ 𝑏 ∩ 𝑐 ∩ 𝑑 = {𝑂} and they are perpendicular two by two. From 𝑑 ⊥ 𝑎, 𝑑 ⊥
𝑎, 𝑑 ⊥ 𝑐 ⟹ 𝑎, 𝑏, 𝑐 are coplanar and 𝑏 ⊥ 𝑎, 𝑐 ⊥ 𝑎, so we can draw to point 𝑂 two
distinct perpendicular lines. False. So the 4 lines cannot be perpendicular two by
two.
We assume that 𝑑 ⊥ 𝛼.
In 𝑑′ ∩ 𝛼 = {𝑂} we draw line 𝑑′′ ⊥ 𝛼. Lines 𝑑′ and 𝑑′′ are concurrent and determine a
plane 𝛽 = (𝑑′, 𝑑′′) and as 𝑂′ ∈ 𝛽, 𝑂′ ∈ 𝛼 ⟹
From (1) and (2) ⟹ in plane 𝛽, on line 𝑎, at point 𝑂′ two distinct perpendicular lines
had been drawn. False. So 𝑑′||𝛼.

202
Reductio ad absurdum. Let 𝑑 ∦ 𝑑′. We draw 𝑑′′||𝑑 through 𝑂’.
⟹ at point 𝑂′ we can draw two perpendicular lines to plane 𝛼. False.
So 𝑑||𝑑′.
Let 𝑑 ⊥ 𝛼 and 𝑑 ∩ 𝛼 = {𝑂}. We draw through 𝑂 a parallel to 𝑑′, which will be
contained in 𝛼, then 𝑑||𝛼.
We assume 𝛽 ⟹ 𝛼 ∩ 𝛽 ≠ ∅ and let 𝐴 ∈ 𝛼 ∩ 𝛽 ⟹ through a point 𝐴 there can be
drawn two distinct perpendicular planes on this line. False.
⟹ 𝛼 || 𝛽.

203
Let 𝑀 be a point in space with the property ||𝑀𝐴|| = ||𝑀𝐵||.
We connect 𝑀 with the midpoint of segment [𝐴𝐵], point 𝑂.
So 𝑀 is on a line drawn through 𝑂, perpendicular to 𝐴𝐵.
But the union of all perpendicular lines drawn through 𝑂 to 𝐴𝐵 is the perpendicular
plane to 𝐴𝐵 at point 𝑂, marked with 𝛼, so 𝑀 ∈ 𝛼.
Vice-versa: let 𝑀 ∈ 𝛼,
Let 𝑀 be a point in space with this property:
Let 𝑂 be the center of the circumscribed circle ∆𝐴𝐵𝐶 ⟹ ||𝑂𝐴|| = 𝑂𝐵|| = ||𝑂𝐶||, so
𝑂 is also a point of the desired locus.
common side

204
According to the previous problem the locus of the points in space equally distant
from 𝐴 and 𝐵 is in the mediator plane of segment [𝐴𝐵], which also contains 𝑀. We
mark with 𝛼 this plane. The locus of the points in space equally distant from 𝐵 and
𝐶 is in the mediator plane of segment [𝐵𝐶], marked 𝛽, which contains both 𝑂 and
𝑀. So 𝛼 ∩ 𝛽 = 𝑂𝑀.
so 𝑀 ∈ the perpendicular line to plane (𝐴𝐵𝐶) in the center of the circumscribed
circle ∆𝐴𝐵𝐶.
Vice-versa, let 𝑀 ∈ this perpendicular line
= ||𝐶𝑀||, so 𝑀 has the property from the statement.
We draw ⊥ from 𝐵 to the plane. Let 𝑂 be the foot of this perpendicular line.
Let
the circle of radius 𝑂𝐴. Vice-versa, let 𝑀 ∈ this circle
so 𝑀 represents the foot from 𝐵 to 𝐴𝑀.

205
Let 𝛼 be a plane that passes through 𝑎 and let 𝑀 be the ⊥ foot from 𝐴 to 𝛼 ⟹
𝐴𝑀 ⊥ 𝛼.
From
so 𝑀 ∈ a perpendicular line to 𝑎 in 𝐴′, thus it is an element of the perpendicular
plane to 𝑎 in 𝐴′, which we mark as 𝜋 and which also contains 𝐴.
𝑀 ∈ the circle of radius 𝐴𝐴′ from plane π.
*Vice-versa, let 𝑀 be a point on this circle of radius 𝐴𝐴′ from plane 𝜋.
⟹ 𝑀 is the foot of a ⊥ drawn from 𝐴 to a plane that passes through 𝑎.
Let 𝐴′ and 𝐵′ be the feet of the perpendicular lines from 𝐴 and 𝐵 to 𝛼
(∃) a plane 𝛽 = (𝐴𝐴′, 𝐵𝐵′) and 𝐴𝐵 ⊂ 𝛽

206
are collinear.
In plane 𝛽 we have
Let 𝑑, 𝑑′ be given lines, 𝐴 given point. We draw through 𝐴 plane 𝛼 ⊥ to 𝑑′.
If 𝑎 ∩ 𝛼 = {𝐵}, then line 𝐴𝐵 is the desired one, because it passes through 𝐴, meets 𝑑
and from 𝑑′
𝛼 𝑑′
𝐴𝐵. If 𝑑 ∩ 𝛼 = ∅ there is no solution.
If 𝑑 ⊂ 𝛼, then any line determined by 𝐴 and a point of 𝑑 represents solution to the
problem, so there are infinite solutions.
right

207
Let 𝑀 be a point such that ||𝐴𝑀|| = 𝑘.
We draw 𝐴𝐴′ ⊥ 𝛼 ⇒ 𝐴′ fixed point and 𝐴𝐴′ ⊥ 𝐴′𝑀.
We write ||𝐴𝐴′|| = 𝑎.
Then
𝑀 ∈ a circle centered at 𝐴′ and of radius √𝑘2 − 𝑎2, for 𝑘 > 𝑎.
For 𝑘 = 𝑎 we obtain 1 point.
For 𝑘 < 𝑎 empty set.
Vice-versa, let 𝑀 be a point on this circle ⟹
so 𝑀 has the property from the statement.

208
In ∆𝑂𝐶𝑀:
But
3. Let 𝐻 be the projection of 𝑂 lcp. plane 𝐴𝐵𝐶, so
𝐻 ∈ corresponding heights of side 𝐴𝐵. We show in the same way that 𝐴𝐶 ⊥
𝐵𝐻 and thus 𝐻 is the point of intersection of the heights, thus orthocenter.

209
First we prove that if a line is ⊥ to two concurrent planes ⟹ the planes coincide.
Let
∆𝐴𝐵𝑀 has two right angles. False. We return to the given problem.
being concurrent, they determine a plane ⟹ 𝐶𝐷 ⊥ 𝐴𝐵.
𝐶, 𝐷, 𝐶′, 𝐷′ are coplanar ⟹ 𝐶𝐶′ and 𝐷𝐷′ are coplanar.

210
We draw
⟹ 𝐵𝐴′ height in ∆𝐵𝐶𝐷 (1)
⟹ 𝐴′𝐶 height in ∆𝐴𝐵𝐶 (2)
From (1) and (2) ⟹ 𝐴′ is the orthocenter ∆𝐴𝐵𝐶 ⟹ 𝑂𝐴′ ⊥ 𝐵𝐶.
Let
∆𝐵𝑂𝑂′ and ∆𝐶𝑂𝑂′ are right at 𝑂′.

211
As
|𝑂𝐴| ≡ |𝑂𝐵| ≡ |𝑂𝐶|
|𝑂𝑂′| common side
}
is the center of the circumscribed circle ∆𝐴𝐵𝐶.
Let 𝐷 be the midpoint of [𝐵𝐶] and 𝐸 ∈ |𝐷𝐴′ such that ||𝐷𝐸|| = ||𝐷𝐴||.
𝐴𝐷 is median in the ∆ isosceles
being external for
Let 𝑀 be a point in the plane and |𝐴𝑀′ the opposite ray to 𝐴𝑀.
According to theorem 1
common

212
We construct 𝐵 on the plane
⟹ 𝐴𝐶 and 𝐵𝐵′ determine a plane 𝛽 = (𝐴𝐶, 𝐵𝐵′) ⟹ 𝐴𝐵 ⊂ 𝛽 and on this plane
𝑀(𝐶𝐴𝐵̂) = 900
− 𝑚(𝐵𝐴𝐵′)̂ .
Let ray |𝐴𝐵 ⊂ 𝛽′ such that 𝐴𝐵 ⊥ 𝑚. Let |𝐴𝐶 another ray such that |𝐴𝐶 ⊂ 𝛽’. We draw
𝐵𝐵′ ⊥ 𝛼 and 𝐶𝐶′ ⊥ 𝛼 to obtain the angle of the 2 rays with 𝛼, namely 𝐵𝐴𝐵′̂ > 𝐶𝐴𝐶′̂.
We draw line |𝐴𝐴’ such that 𝐴𝐴′ ⊥ 𝛼 and is on the same side of plane 𝛼 as well as
half-plane 𝛽’.
[𝐴𝐵 is the projection of ray [𝐴𝐴′ on plane 𝛽
Let 𝑑 be the border of 𝛼′ and 𝐴 ∈ 𝑑. We draw a plane ⊥ on 𝑑 in 𝐴, which we mark
as 𝛾.

213
In this plane, there is only one ray 𝑏, with its origin in 𝐴, such that 𝑚(𝑐, 𝑏̂ ) = 𝑎.
The desired half-plane is determined by 𝑑 and ray 𝑏, because from
Let 𝑑 be the edge of the dihedral angle and 𝐴 ∈ 𝑑. We draw 𝑎 ⊥ 𝑑, 𝑎 ⊂ 𝛼′ and 𝑏 ⊥
𝑑, 𝑏 ⊂ 𝛽’ two rays with origin in 𝐴. It results 𝑑 ⊥ (𝑎𝑏). We draw on plane (𝑎, 𝑏) ray 𝑐
such that 𝑚(𝑎𝑐̂) = 𝑚(𝑐𝑏̂) (1).
As 𝑑 ⊥ (𝑎𝑏) ⟹ 𝑑 ⊥ 𝑐.
Half-plane 𝛾′ = (𝑑, 𝑐) is the desired one, because
If we consider the opposite ray to 𝑐, 𝑐′, half-plane 𝛾′′ = (𝑑, 𝑐′) also forms
concurrent angles with the two half-planes, being supplementary to the others.

214
Let 𝑀 be a point in space equally distant from the half-planes 𝛼′, 𝛽′ ⟹ ||𝑀𝐴|| =
||𝑀𝐵||.
where 𝑑 = 𝛼 ∩ 𝛽.
Let
|𝑀𝐴| = |𝑀𝐵|
|𝑂𝑀| common side right triangle
} ⟹
⟹ 𝑀 ∈ bisector of the angle.
𝐴𝐷𝐵̂ ⟹ 𝑀 ∈ bisector half-plane of the angle of half-planes 𝛼′, 𝛽′.
If 𝑀′ is equally distant from half-planes 𝛽′ and 𝛼′′ we will show in the same way
that 𝑀′ ∈ bisector half-plane of these half-planes. We assume that 𝑀 and 𝑀′ are on
this plane ⊥ to 𝑑, we remark that 𝑚(𝑀𝑂𝑀′̂ ) = 900
, so the two half-planes are ⊥.
Considering the two other dihedral angles, we obtain 2 perpendicular planes, the 2
bisector planes.
Vice-versa: we can easily show that a point on these planes is equally distant form
planes 𝛼 and 𝛽.

215
Let 𝛼 ∩ 𝛽 = 𝛼. In plane 𝛽 we draw
As
but
so from a point it can be drawn only one perpendicular line to a plane,
Let

216
the line with the same direction. We know that the union of the lines with the same
direction and are based on a given line is a plane. As this plane contains a
perpendicular line to 𝛼, it is perpendicular to 𝛼.
Let 𝛼 = (𝑑1, 𝑑2) the plane of the two concurrent lines and 𝑀 is a point with the
property 𝑑(𝑀, 𝑑1) = 𝑑(𝑀, 𝑑2). We draw
Let
a bisector of the angle formed by the two lines, and 𝑀 is on a line 𝛼 which
meets a bisector ⇒ 𝑀 ∈ a plane ⊥ 𝛼 and which intersects 𝛼 after a bisector. Thus
the locus will be formed by two planes ⊥ α and which intersects 𝛼 after the two
bisectors of the angle formed by 𝑑1, 𝑑2. The two planes are ⊥.
⟹
‖𝑀′𝐴‖ = ‖𝑀′𝐵‖
||MM′|| common side
} ⟹ 𝑀𝐴 ⊥ 𝑑1
And in the same way 𝑀𝐵 ⊥ 𝑑2 ⇒ 𝑀 has the property from the statement.

217
Let 𝛽 ∩ 𝛾 = 𝑑 and 𝑀 ∈ 𝑑 ⟹ 𝑀 ∈ 𝛽, 𝑀 ∈ 𝛾. We draw ⊥ from 𝑀 to 𝛼, line 𝑑’.
According to a previous problem
Let 𝛽 and 𝛾 be such planes, that is
From
are secant planes and ⊥ to 𝛼.
So their intersection is ⊥ through 𝐴 to plane 𝛼.
We construct the point on the two planes and the desired plane is determined by
the two perpendicular lines.
Let 𝛼 ∩ 𝛽 = 𝑑 and 𝑀 ∈ 𝑑. We consider a ray originating in 𝑀, 𝑎 ∈ 𝛼 and we
construct a ⊥ plane to 𝑎 in 𝑀, plane 𝛾.
Because

218
and let a ray originating in 𝑀, 𝑏 ⊂ 𝛽 ∩ 𝛾 ⇒ 𝑏 ⊂ 𝛽 , 𝑏 ⊂.𝛾 As 𝑎 ⊥ 𝛾 ⟹ 𝑎 ⊥ 𝑏 and the
desired plane is that determined by rays (𝑎, 𝑏).
Let 𝛼 ∩ 𝛽 = 𝑎 and 𝑑 ∩ 𝛽 = {𝐴}.
We suppose that 𝐴 ∉ 𝑎. Let 𝑀 ∈ 𝑎, we build 𝑏 ⊥ 𝛽, 𝑀 ∈ 𝑏 ⟹ 𝑏 ⊂ 𝛼.
If

219
From (1), (2), (3) ⟹ 𝑎 ∥ 𝑏 ∥ 𝑐.
Let 𝐴 ∈ int. (𝛼′ 𝛽′̂), 𝛼 ∩ 𝛽 = 𝑑.
Let 𝐴 ∈ int. (𝛼′′ 𝛽′̂ ). We show the same way that
𝑚(𝐵𝐴𝐶̂) = 1800
− 𝑚(𝛼′′ 𝛽′̂ )
𝑚(𝛼′′ 𝛽′̂ ) = 1800
− 𝑚(𝛼′ 𝛽′̂)
} ⟹ 𝑚(𝐵𝐴𝐶̂) = 1800
− 1800
+ 𝑚(𝛼′ 𝛽′̂) = 𝑚(𝛼′ 𝛽′̂).
If 𝐴 ∈ int. (𝛼′′ 𝛽′′̂ ) ⟹ 𝑚(𝐵𝐴𝐶̂) = 1800
− 𝑚(𝛼′′ 𝛽′̂ ).
If 𝐴 ∈ int. (𝛼′ 𝛽′′̂ ) ⟹ 𝑚(𝐵𝐴𝐶̂) = 1800
− 𝑚(𝛼′ 𝛽′̂).

220
This book is a translation from Romanian of "Probleme Compilate şi Rezolvate de Geometrie şi
Trigonometrie" (University of Kishinev Press, Kishinev, 169 p., 1998), and includes problems of 2D
and 3D Euclidean geometry plus trigonometry, compiled and solved from the Romanian Textbooks
for 9th and 10th grade students, in the period 1981-1988, when I was a professor of mathematics
at the "Petrache Poenaru" National College in Balcesti, Valcea (Romania), Lycée Sidi El Hassan
Lyoussi in Sefrou (Morocco), then at the "Nicolae Balcescu" National College in Craiova and
Dragotesti General School (Romania), but also I did intensive private tutoring for students
preparing their university entrance examination. After that, I have escaped in Turkey in September
1988 and lived in a political refugee camp in Istanbul and Ankara, and in March 1990 I immigrated
to United States. The degree of difficulties of the problems is from easy and medium to hard. The
solutions of the problems are at the end of each chapter. One can navigate back and forth from
the text of the problem to its solution using bookmarks. The book is especially a didactical
material for the mathematical students and instructors.
The Author

Compiled and solved problems in geometry and trigonometry,F.Smaradanhe

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