IMO 1963 - Problem 7

Problem: Prove that a tetrahedron $SABC$ has five different spheres that touch all six lines determined by its edges if and only if it is regular.
Solution: The sphere are arranged in a similar manner as in the planar case where we have one incircle and three excircles. Here we have one "insphere" and four "exspheres" corresponding to each of the four sides. Each vertex of the tetrahedron effectively has three tangent lines drawn from it to each of the five spheres. Repeatedly using the equality of the three tangent segments from a vertex (in the same vein as for tangent planar quadrilaterals ) we obtain $SA+BC=SB+CA=SC+AB$ from the insphere. From the hence $SA=SB=SC$ and $AB=BC=CA$. By symmetry, we also have $AB=AC=AS$. Hence indeed, all the edges of the tetraghedron are equal in length and thus we have shown that the tetrahedron is regular.