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