PL-OM 2006 Problem 5

Tetrahedron $ABCD$ in which $AB=CD$ is given. Sphere inscribed in it is tangent to faces $ABC$ and $ABD$ respectively in $K$ and $L$ . Prove that if points $K$ and $L$ are centroids of faces $ABC$ and $ABD$ then tetrahedron $ABCD$ is regular.