Question 99
Let $\odot I$ be the incircle of $\triangle ABC$. The circle $\odot I$ intersects sides $AB$, $BC$ and $CA$ at points $D$, $E$ and $F$, respectively. Line $EF$ intersects lines $AI$, $BI$ and $DI$ at points $M$, $N$ and $K$, respectively. Prove that $DM \cdot KE = DN \cdot KF$. (posed by Zhang Pengcheng)