Question 42
In a triangle $ABC$ with incenter $I$, the incircle of $ABC$ touches the side $[BC]$ at the point $D$ and let $T$ be the midpoint of the line segment $[ID]$. The line passing through $I$ and perpendicular to $AD$ intersects the lines $AB$ and $AC$ at the points $K$ and $L$, respectively. The line passing through $T$ and perpendicular to $AD$ intersects the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Show that $KM \cdot LN = BM \cdot CN$.
