ELMO SL 2013 G1

Let $ABC$ be a triangle with incenter $I$ . Let $U$ , $V$ and $W$ be the intersections of the angle bisectors of angles $A$ , $B$ , and $C$ with the incircle, so that $V$ lies between $B$ and $I$ , and similarly with $U$ and $W$ . Let $X$ , $Y$ , and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$ , $AC$ , and $AB$ , respectively. Let triangle $UVW$ be the David Yang triangle of $ABC$ and let $XYZ$ be the Scott Wu triangle of $ABC$ . Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral.

Proposed by Owen Goff