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

## Own Problem involving regular polygons

Suppose is a pyramid whose base is a cyclic polygon, and denote the intersection of the edges with a plane . Prove that if the polygon is regular, then the polygon 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 lie on a plane which intersects the cone by a conic section. Since five points determine uniquely the conic, and the points are on a circle, it follows that the plane intersects the cone by a circle, and is therefore parallel with the base. Then can be obtained from by a certain homothety of center . This means that is also regular.

Advertisements

Categories: Affine Geometry, Geometry, Olympiad, Problem Solving
Geometry

Comments (0)
Trackbacks (0)
Leave a comment
Trackback