Prove that if two segments of a tetrahedron, going from the ends of some edge to the centers of the inscribed circles of opposite faces, intersect, then the segments issued from the ends of the crossing with it edges to the centers of the inscribed circles of the other two faces, also intersect.
Problem
Source: All-Russian MO 2001 Regional (R4) 11.6
Tags: geometry, concurrency, concurrent, tetrahedron, 3D geometry