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.