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Sphere touches all edges of an octahedron STABCD. Prove that ABCD are complanar

Source: 2021 Oral Moscow Geometry Olympiad grades 10-11 p4

May 17, 2021
geometry3D geometryoctahedronsphere

Problem Statement

Points STABCDSTABCD in space form a convex octahedron with faces SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDASAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA such that there exists a sphere that is tangent to all of its edges. Prove that A,B,C,DA,B,C,D lie in one plane.
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