Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle, and
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
the midpoint of its side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
. Let
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
be the incircle of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. The median
![AM](/media/m/9/2/1/921d54bb92ada2d2120b2591b722ea12.png)
of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
intersects the incircle
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
at two points
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
. Let the lines passing through
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
and
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
, parallel to
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
, intersect the incircle
![\gamma](/media/m/2/4/a/24aca7af13a8211060a900a49ef999e9.png)
again in two points
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
. Let the lines
![AX](/media/m/3/a/8/3a8b3cfe621304b5621fb712075419c2.png)
and
![AY](/media/m/b/b/9/bb90496e08c85f2dceaaf0a186d021fe.png)
intersect
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
again at the points
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
. Prove that
![BP = CQ](/media/m/f/b/9/fb9909fd1cf9ceb5780c25a9187dc6e0.png)
.
%V0
Let $ABC$ be a triangle, and $M$ the midpoint of its side $BC$. Let $\gamma$ be the incircle of triangle $ABC$. The median $AM$ of triangle $ABC$ intersects the incircle $\gamma$ at two points $K$ and $L$. Let the lines passing through $K$ and $L$, parallel to $BC$, intersect the incircle $\gamma$ again in two points $X$ and $Y$. Let the lines $AX$ and $AY$ intersect $BC$ again at the points $P$ and $Q$. Prove that $BP = CQ$.