Question 40
Let $K$ and $L$ be points on side $BC$, $M$ a point on side $AC$, and $N$ a point on side $AB$ of triangle $ABC$. These points are chosen such that $\triangle KMC \sim \triangle ABC$ and $\triangle LBN \sim \triangle ABC$. Suppose that the segments $NL$ and $KM$ intersect at point $P$ inside triangle $ABC$. The circumcircle of triangle $AMP$ intersects the line $CP$ again at point $X$, and the circumcircle of triangle $ANP$ intersects the line $BP$ again at point $Y$. Prove that the points $B, C, X$, and $Y$ lie on the same circle.