Problems

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.

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.

Prove that a circle under the axial symmetry transforms into a circle.

A quadrilateral has an axis of symmetry. Prove that this quadrilateral is either an isosceles trapezoid or is symmetric with respect to its diagonal.

The symmetry axis of the polygon intersects its sides at points $A$ and $B$. Prove that the point $A$ is either the vertex of the polygon or the middle of the side perpendicular to the axis of symmetry.

Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.