Question 178
Let $ABCD$ be a triangular pyramid. A sphere $\omega_A$ is tangent to the face $BCD$, and to the planes of the other faces outside the faces. Similarly, a sphere $\omega_B$ is tangent to the face $ACD$, and to the planes of the other faces outside the faces. Let $\omega_A$ meet the plane $ACD$ at $K$, and let $\omega_B$ meet the plane $BCD$ at $L$. The points $X$ and $Y$ are chosen on the extensions of the segments $AK$ and $BL$ beyond $K$ and $L$, respectively, so that $\angle CKD = \angle CXD + \angle CBD$ and $\angle CLD = \angle CYD + \angle CAD$. Prove that the points $X$ and $Y$ are equidistant from the midpoint of $CD$.