Problem

Source: Sharygin Geometry Olympiad 2016 First Round P23 grades 10-11

Tags: geometry, 3D geometry, sphere



A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle? (It is not obligatory to use all segments. The side of the triangle can be formed by two segments)