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A_1X / A_1O +B_1X/B_1O +C_1X/C_1O +D_1X/D_1O =4, insphere of tetrahedron

Source: I Soros Olympiad 1994-95 Ukraine R2 11.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 6, 2024
geometry3D geometryinspheretetrahedron

Problem Statement

A tetrahedron ABCDABCD is given, in which each pair of adjacent edges are equal segments. Let OO be the center of the sphere inscribed in this tetrahedron . XX is an arbitrary point inside the tetrahedron, X≠OX \ne O. The line OXOX intersects the planes of the faces of the tetrahedron at the points marked by A1A_1, B1B_1, C1C_1, D1D_1. Prove that A1XA1O+B1XB1O+C1XC1O+D1XD1O=4\frac{A_1X}{A_1O} +\frac{B_1X}{B_1O} +\frac{C_1X}{C_1O}+\frac{D_1X}{D_1O}=4
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