IMO Shortlist 1962 problem 7
The tetrahedron
has the following property: there exist five spheres, each tangent to the edges
or to their extensions.
a) Prove that the tetrahedron
is regular.
b) Prove conversely that for every regular tetrahedron five such spheres exist.
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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.
Source: Međunarodna matematička olimpijada, shortlist 1962
Školjka 
