Home > Affine Geometry, Geometry, Olympiad, Problem Solving > Own Problem involving regular polygons

## Own Problem involving regular polygons

Suppose $VA_1A_2...A_n,\ n \geq 5$ is a pyramid whose base is a cyclic polygon, and denote $B_1B_2..B_n$ the intersection of the edges $VA_1,VA_2,...,VA_n$ with a plane $\alpha$. Prove that if the polygon $B_1B_2...B_n$ is regular, then the polygon $A_1A_2...A_n$ is also regular.

Beniamin Bogosel, Shortlist for Romanian National Olympiad

Solution: Because the base is a cyclic polygon, the pyramid can be inscribed in a cone, and $B_1,..,B_n$ lie on a plane which intersects the cone by a conic section. Since five points determine uniquely the conic, and the points $B_k$ are on a circle, it follows that the plane $\alpha$ intersects the cone by a circle, and is therefore parallel with the base. Then $A_1...A_n$ can be obtained from $B_1...B_n$ by a certain homothety of center $V$. This means that $A_1...A_n$ is also regular.