Question 12
Let $ABC$ be an acute, non-isosceles triangle with the circumcircle ($O$). Denote $D, E$ as the midpoints of $AB, AC$ respectively. Two circles $(ABE)$ and $(ACD)$ intersect at $K$ differs from $A$. Suppose that the ray $AK$ intersects ($O$) at $L$. The line $LB$ meets $(ABE)$ at the second point $M$ and the line $LC$ meets $(ACD)$ at the second point $N$.
1. Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$.
2. Prove that $K$ is the midpoint of $MN$.
1. Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$.
2. Prove that $K$ is the midpoint of $MN$.