Question 9
Let $D$ be a point on the side $BC$ of triangle $ABC$ such that $AB + BD = AC + CD$. The line segment $AD$ cuts the incircle of triangle $ABC$ at $X$ and $Y$ with $X$ closer to $A$. Let $E$ be the point of contact of the incircle of triangle $ABC$ on the side $BC$. Show that
(i) $EY$ is perpendicular to $AD$,
(ii) $XD$ is $2IA'$, where $I$ is the incentre of the triangle $ABC$ and $A'$ is the midpoint of $BC$.
(i) $EY$ is perpendicular to $AD$,
(ii) $XD$ is $2IA'$, where $I$ is the incentre of the triangle $ABC$ and $A'$ is the midpoint of $BC$.