Problems

Find all triangles in which the angles form an arithmetic progression, and the sides form: a) an arithmetic progression; b) a geometric progression.

Prove that the point $X$ lies on the line $AB$ if and only if $\overrightarrow{OX}=t\overrightarrow{OA}+(1-t)\overrightarrow{OB}$ for some $t$ and any point $O$.

Several points are given and for some pairs $(A,B)$ of these points the vectors $\overrightarrow{AB}$ are taken, and at each point the same number of vectors begin and end. Prove that the sum of all the chosen vectors is $\overrightarrow{0}$.

Prove that the medians of the triangle $ABC$ intersect at one point and that point divides the medians in a ratio of $2:1$, counting from the vertex.

Prove that, when a circle is translated it becomes a circle.

Two circles of radius $R$ touch at point $E$. On one of them, point $B$ is chosen and on the other point $D$ is chosen. These points have a property of $\mathrm{\angle }BED={90}^{\circ }$. Prove that $BD=2R$.

Two circles of radius $R$ intersect at points $D$ and $B$. Let $F$ and $G$ be the points of intersection of the middle perpendicular to the segment $BD$ with these circles lying on one side of the line $BD$. Prove that $B{D}^2+F{G}^2=4R^2$.

Inside the rectangle $ABCD$, the point $E$ is taken. Prove that there exists a convex quadrilateral with perpendicular diagonals of lengths $AB$ and $BC$ whose sides are equal to $AE$, $BE$, $CE$, $DE$.

Prove that, with central symmetry, a circle transforms into a circle.

The opposite sides of a convex hexagon are pairwise equal and parallel. Prove that it has a centre of symmetry.