Question 8
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C \lt \angle A \lt 90^{\circ}$ in such a way that $BD = BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incentre of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
(Russia)
(Russia)