RU-ALLRUSSIAN 1994 Problem 117

The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$ . Prove that the tetrahedron $ABCD$ is regular.

(D. Tereshin)