Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$.

In the triangle $ABC$ with sides $BC> AC> AB$ the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three. (Rozhkova Maria)

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

A triangle $ABC$ satisfies $BC = AC +\frac12 AB$. Point $P$ on side $AB$ is taken so that $AP = 3PB$. Prove that $ \angle PAC = 2\angle CPA$.

Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties: [list] [*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$. [*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$. [/list]

Let $ABCD$ be a rectangle. The point $E$ lies on side $ \overline{AB}$ and the point $F$ is lies side $ \overline{AD}$, such that $\angle FEC=\angle CEB$ and $\angle DFC=\angle CFE$. Determine the measure of the angle $\angle FCE$ and the ratio $AD/AB$.

Let $ABC$ be a triangle with an interior point $P$ such that $\angle APB = \angle APC = \alpha$ and $\alpha > 180^o-\angle BAC$. The circumcircle of triangle $APB$ cuts $AC$ at $E$, the circumcircle of triangle $APC$ cuts $AB$ at $F$. Let $Q$ be the point in the triangle $AEF$ such that $\angle AQE = \angle AQF =\alpha$. Let $D$ be the symmetric point of $Q$ wrt $EF$. Angle bisector of $\angle EDF$ cuts $AP$ at $T$. a) Prove that $\angle DET = \angle ABC, \angle DFT = \angle ACB$. b) Straight line $PA$ cuts straight lines $DE, DF$ at $M, N$ respectively. Denote $I, J$ the incenters of the triangles $PEM, PFN$, and $K$ the circumcenter of the triangle $DIJ$. Straight line $DT$ cut $(K)$ at $H$. Prove that $HK$ passes through the incenter of the triangle $DMN$.

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Given an isosceles triangle $BAC$ with vertex angle $\angle BAC =20^o$. Construct an equilateral triangle $BDC$ such that $D,A$ are on the same side wrt $BC$. Construct an isosceles triangle $DEB$ with vertex angle $\angle EDB = 80^o$ and $C,E$ are on the different sides wrt $DB$. Prove that the triangle $AEC$ is isosceles at $E$.

Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.

Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .

Let $ABC$ be an acute-angled triangle in which $\angle BAC = 60^o$ and $AB> AC$. Let $H$ and $I$ denote the points of intersection of the altitudes and angle bisectors of this triangle, respectively. Find the ratio $\angle ABC: \angle AHI$.

Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$

Two of the Geometry box tools are placed on the table as shown. Determine the angle $\angle ABC$ [img]https://2.bp.blogspot.com/--DWVwVQJgMM/XU1OK08PSUI/AAAAAAAAKfs/dgZeYwiYOrQJE4eKQT5s13GQdBEHPqy9QCK4BGAYYCw/s1600/prmo%2B16%2BChandigarh%2Bp7.png[/img]

Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$. Ya. Konstantinovski

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB' $meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$. (D. Krekov)

The straight line containing the centers of the circumscribed and inscribed circles of triangle $ABC$ intersects rays $BA$ and $BC$ and forms an angle with the altitude to side $BC$ equal to half the angle $\angle BAC$. What is angle $\angle ABC$?

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.