P23

Part of 2016 Sharygin Geometry Olympiad

Problems(1)

can segments of tangents of sphere circumscribed on 4dron, form triangle?

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

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)

geometry3D geometrysphere