KöMaL 2019 B5053

Let $G$ denote the inscribed sphere of a tetrahedron $ABCD$, and let $G_A$ be the escribed sphere touching the face $BCD$. Let $T$ be the tetrahedron formed by the points of tangency of $G$ on the planes of the faces, and let $T_A$ be the tetrahedron formed by the points of tangency of $G_A$ on the planes of the faces. Show that $\frac{V^3(T)}{V^3(T_A)} =\frac{V^2(G)}{V^2(G_A)}$ holds for the volumes of the tetrahedra and the spheres.