Question 137
Let $D$, $E$, $F$ denote the tangent points of the incircle of $ABC$ with the sides $BC$, $AC$, $AB$, respectively. Let $M$ be the midpoint of the segment $EF$. Let $L$ be the intersection point of the circle passing through $D$, $M$, $F$ and the segment $AB$, $K$ be the intersection point of the circle passing through $D$, $M$, $E$ and the segment $AC$.
Prove that the circle passing through $A$, $K$, $L$ touches the line $BC$.
(V. Voinov)
Prove that the circle passing through $A$, $K$, $L$ touches the line $BC$.
(V. Voinov)