Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular. Freddie Illingworth & Dominic Yeo