IL-GILLIS 2015 Problem 5

Let $ABCD$ be a tetrahedron. Denote by $S_1$ the inscribed sphere inside it, which is tangent to all four faces. Denote by $S_2$ the outer escribed sphere outside $ABC$ , tangent to face $ABC$ and to the planes containing faces $ABD,ACD,BCD$ . Let $K$ be the tangency point of $S_1$ to the face $ABC$ , and let $L$ be the tangency point of $S_2$ to the face $ABC$ . Let $T$ be the foot of the perpendicular from $D$ to the face $ABC$ .

Prove that $L,T,K$ lie on one line.