Problem 1893

The convex polyhedron $\mathcal{K}$ has five vertices, $A$, $B$, $C$, $D$ and $E$. The line segment $DE$ intersects the plane of the triangle $ABC$ in the interior of the triangle. Show that $\mathcal{K}$ has an inscribed sphere – with being tangent to all six faces – if and only if the inscribed spheres of the terahedra $ABCD$ and $ABCE$ are tangent to each other.