Question 27
Let $ABC$ be a non-isosceles triangle, and let $\Gamma$ be its in-circle. Let $D$, $E$, $F$ be the points of contact of $\Gamma$ with the sides $BC$, $CA$, $AB$ respectively. Suppose $FD$, $DE$, $EF$ intersect $CA$, $AB$, $BC$ in $U$, $V$, $W$ respectively. If $L$, $M$, $N$ are respectively the mid-points of $DW$, $EU$, $FV$ prove that $L$, $M$, $N$ are collinear.