Question 99
Let $ABC$ be a triangle with $\angle A = 90^{\circ}$ and $AB \lt AC$. Let $AD$ be the altitude from $A$ to $BC$. Let $P$, $Q$ and $I$ denote respectively the incentres of triangles $ABD$, $ACD$ and $ABC$. Prove that $AI$ is perpendicular to $PQ$ and $AI = PQ$.