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The relation between area of a polygon and the pyramid

Source: IMO LongList 1982 - P13

March 16, 2011
geometry3D geometrypyramidspheretrigonometrycircumcircletrapezoid

Problem Statement

A regular nn-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by S1,S2S_1, S_2, and SS, respectively. Let σ\sigma be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that σS=4S1S2cos⁡2πn.\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.
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