Let

be a triangle, and

the midpoint of its side

. Let

be the incircle of triangle

. The median

of triangle

intersects the incircle

at two points

and

. Let the lines passing through

and

, parallel to

, intersect the incircle

again in two points

and

. Let the lines

and

intersect

again at the points

and

. Prove that

.
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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 $BP = CQ$.