Školjka

%V0 Let $ABC$ be a triangle such that $\widehat{ACB} < \widehat{BAC} < \frac {\pi}{2}$. Let $D$ be a point of $[AC]$ such that $BD = BA$. The incircle of $ABC$ touches $[AB]$ at $K$ and $[AC]$ at $L$. Let $J$ be the center of the incircle of $BCD$. Prove that $(KL)$ intersects $[AJ]$ at its middle.