Problem

Source: IMO 1962, Day 2, Problem 7

Tags: geometry, 3D geometry, tetrahedron, sphere, incenter, IMO, IMO 1962



The tetrahedron $SABC$ has the following property: there exist five spheres, each tangent to the edges $SA, SB, SC, BC, CA, AB,$ or to their extensions. a) Prove that the tetrahedron $SABC$ is regular. b) Prove conversely that for every regular tetrahedron five such spheres exist.