Question 21
The incircle of a triangle $ABC$ is tangent to $BC$, $AC$, $AB$ at the points $D$, $E$, $F$, respectively. Suppose the line $EF$ intersects the lines $BI$, $CI$, $BC$, $DI$ at the points $K$, $L$, $M$, $Q$, respectively, where the incenter of $\triangle ABC$ is $I$. If the line passing through both the midpoint of $CL$ and $M$ intersects $CK$ at a point $P$, show that
$$
PQ = \frac{AB \cdot KQ}{BI}.
$$
$$
PQ = \frac{AB \cdot KQ}{BI}.
$$