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sum sin^2 (<A_iMA_{i+1}) / d(M,A_iA_{i+1})= ... in tetrahedron

Source: 1998 Romania NMO IX p4

August 14, 2024
geometrytetrahedrontrigonometry3D geometry

Problem Statement

Let A1A2...AnA_1A_2...A_n be a regular polygon (n>4n > 4), TT be the common point of A1A2A_1A_2 and An−1AnA_{n-1}A_n and MM be a point in the interior of the triangle A1AnTA_1A_nT. Show that the equality ∑i=1n−1sin⁡2(∠AiMAi+1)d(M,AiAi+1=sin⁡2(∠A1MAn)d(M,A1An\sum_{i=1}^{n-1} \frac{\sin^2 \left(\angle A_iMA_{i+1}\right)}{d(M,A_iA_{i+1}}=\frac{\sin^2 \left(\angle A_1MA_n\right)}{d(M,A_1A_n} holds if and only if MM belongs to the circumcircle of the polygon.
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