Question 25
Let $ABC$ be a scalene triangle with angle bisectors $AD$, $BE$, and $CF$ so that $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$ respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$ respectively. Prove that line $AN$ and the line through $M$ parallel to $AD$ intersect on the circumcircle of $ABC$ if and only if $DE = DF$.