Problem

Source: romanian tst 2 - 2007 , problem 2

Tags: geometry, circumcircle, geometry proposed



Let $ABC$ be a triangle, $E$ and $F$ the points where the incircle and $A$-excircle touch $AB$, and $D$ the point on $BC$ such that the triangles $ABD$ and $ACD$ have equal in-radii. The lines $DB$ and $DE$ intersect the circumcircle of triangle $ADF$ again in the points $X$ and $Y$. Prove that $XY\parallel AB$ if and only if $AB=AC$.