Problems

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.

A parallelogram $ABCD$ and a point $E$ are given. Through the points $A,B,C,D$, lines parallel to the straight lines $EC,ED,EA,EB$, respectively, are drawn. Prove that they intersect at one point.

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.

Prove that a circle transforms into a circle when it is rotated.

Prove that a convex $n$-gon is regular if and only if it is transformed into itself when it is rotated through an angle of ${360}^{\circ }/n$ with respect to some point.

Prove that the triangle $ABC$ is regular if and only if, by turning it by ${60}^{\circ }$ (either clockwise or anticlockwise) with respect to point A, its vertex B moves to $C$.