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.
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.
Prove that if a shape has two perpendicular axes of symmetry, then it has a centre of symmetry.
The tower in the castle of King Arthur is crowned with a roof, which is a triangular pyramid, in which all flat angles at the top are straight. Three roof slopes are painted in different colours. The red roof slope is inclined to the horizontal at an angle \(\alpha\), and the blue one at an angle \(\beta\). Find the probability that a raindrop that fell vertically on the roof in a random place fell on the green area.