Question 76
Let $ABC$ be a triangle. Point $D$ lies on side $BC$, such that the incircles of triangles $ABD$ and $ACD$ are congruent. Let $\Omega_B$ be the circle with diameter $AB$, and let $\Omega_C$ be the circle with diameter $AC$. Prove that line $AD$ is perpendicular to one of the common tangents to the circles $\Omega_B$ and $\Omega_C$.