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BC_1 + BC_2 + ... + BC_n is fixed sum, regurar n- sides pyramid

Source: Netherlands - Dutch MO 1989 p4

December 25, 2022
geometryfixed3D geometrypyramid

Problem Statement

Given is a regular nn-sided pyramid with top TT and base A1A2A3...AnA_1A_2A_3... A_n. The line perpendicular to the ground plane through a point BB of the ground plane within A1A2A3...AnA_1A_2A_3... A_n intersects the plane TA1A2TA_1A_2 at C1C_1, the plane TA2A3TA_2A_3 at C2C_2, and so on, and finally the plane TAnA1TA_nA_1 at CnC_n. Prove that BC1+BC2+...+BCnBC_1 + BC_2 + ... + BC_n is independent of choice of BB's.
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