Problem

Source: 2016 ELMO Problem 6

Tags: geometry, circumcircle, angle bisector, incircle, Inversion, tangent circles, Elmo



Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$. (a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$. (b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$. James Lin