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.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.