Question 2
Triangle $ABC$ is right-angled at $A$, with $AB\lt AC$ and altitude $AH$. Let $I,J$ be the incenters of triangles $AHB,AHC$, respectively. Set $D=AI\cap BH$, $F=IJ\cap AC$, and $K=DJ\cap BF$. Prove that the circumcircle of triangle $AKD$ is tangent to $BC$.