Problem

Source: ELMO Shortlist 2013: Problem G1, by Owen Goff

Tags: geometry, incenter, circumcircle, inradius, geometry unsolved



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