Question 5
Let $ABC$ be a triangle with $AB \neq AC$. Its incircle has center $I$ and touches the side $BC$ at point $D$. Line $AI$ intersects the circumcircle $\mathcal{C}$ of triangle $ABC$ at $M$ and $DM$ intersects again $\mathcal{C}$ at $P$. Prove that $\widehat{API}=90^{\circ}$.