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r1 + r2 + r3 + r4 = 2r for radii of inspheres of terrahedron

Source: 1985 German Federal - Bundeswettbewerb Mathematik - BWM - Round 2 p2

November 21, 2022
geometry3D geometrytetrahedroninradiusSpheres

Problem Statement

The insphere of any tetrahedron has radius rr. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are r1,r2,r3r_1, r_2, r_3 and r4r_4. Prove that r1+r2+r3+r4=2rr_1 + r_2 + r_3 + r_4 = 2r
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