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volume of tetrahedron as a function

Source: Yugoslav TST 1997 P1

May 15, 2021
geometry3D geometrytetrahedronfunction

Problem Statement

Consider a regular nn-gon A1A2…AnA_1A_2\ldots A_n with area SS. Let us draw the lines l1,l2,…,lnl_1,l_2,\ldots,l_n perpendicular to the plane of the nn-gon at A1,A2,…,AnA_1,A_2,\ldots,A_n respectively. Points B1,B2,…,BnB_1,B_2,\ldots,B_n are selected on lines l1,l2,…,lnl_1,l_2,\ldots,l_n respectively so that:
(i) B1,B2,…,BnB_1,B_2,\ldots,B_n are all on the same side of the plane of the nn-gon; (ii) Points B1,B2,…,BnB_1,B_2,\ldots,B_n lie on a single plane; (iii) A1B1=h1,A2B2=h2,…,AnBn=hnA_1B_1=h_1,A_2B_2=h_2,\ldots,A_nB_n=h_n.
Express the volume of polyhedron A1A2…AnB1B2…BnA_1A_2\ldots A_nB_1B_2\ldots B_n as a function in S,h1,…,hnS,h_1,\ldots,h_n.
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