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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.

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