Question 80
Circles $k_1$ and $k_2$ meet at points $M$ and $N$. The line $l$ meets the circle $k_1$ in points $A$ and $C$, and the circle $k_2$ in points $B$ and $D$ so that the points $A, B, C$ and $D$ are on the line $l$ in that order. Let $X$ be a point on the line $MN$ such that the point $M$ is between points $X$ and $N$. Let $P$ be the intersection of lines $AX$ and $BM$, and $Q$ the intersection of lines $DX$ and $CM$.
If $K$ is the midpoint of the segment $\overline{AD}$, and $L$ the midpoint of the segment $\overline{BC}$, prove that the lines $XK$ and $ML$ meet on the line $PQ$.
(Matija Bucić)
If $K$ is the midpoint of the segment $\overline{AD}$, and $L$ the midpoint of the segment $\overline{BC}$, prove that the lines $XK$ and $ML$ meet on the line $PQ$.
(Matija Bucić)