Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle such that
![\widehat{ACB} < \widehat{BAC} < \frac {\pi}{2}](/media/m/f/d/5/fd5def1e90f4a92a7aa3f4319effe9ed.png)
. Let
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
be a point of
![[AC]](/media/m/c/7/4/c741f6e67c75b27db5e363fbcf5393c6.png)
such that
![BD = BA](/media/m/7/a/7/7a7e5b9ff262f4a9799d306be5d7666f.png)
. The incircle of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
touches
![[AB]](/media/m/1/0/6/10652e809a6cb10509d19772365232e2.png)
at
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![[AC]](/media/m/c/7/4/c741f6e67c75b27db5e363fbcf5393c6.png)
at
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
. Let
![J](/media/m/9/0/e/90ef5cc2558381e341da5808eb92126f.png)
be the center of the incircle of
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
. Prove that
![(KL)](/media/m/c/2/c/c2c82c50f0a9d1b338b18ba985dff3e2.png)
intersects
![[AJ]](/media/m/4/d/3/4d35637bd1d22b6a0da26de27970690f.png)
at its middle.
%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.