Question 21
Let $ABC$ be a triangle with $AB = AC \neq BC$. Further, let $I$ be the incentre of $ABC$.
The line $BI$ meets $AC$ at the point $D$, and the perpendicular to $AC$ through $D$ meets $AI$ at the point $E$. Prove that the reflection of $I$ in the line $AC$ lies on the circumcircle of triangle $BDE$.
The line $BI$ meets $AC$ at the point $D$, and the perpendicular to $AC$ through $D$ meets $AI$ at the point $E$. Prove that the reflection of $I$ in the line $AC$ lies on the circumcircle of triangle $BDE$.