Let $$ABC$$ be a triangle with $$\angle A = 90^{\circ}$$ and $$AB < AC$$. Let $$AD$$ be the altitude from $$A$$ on 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 $$AI = PQ$$.