The incenter, two arc midpoints and ML ⊥ NK
Problem
Let $ABC$ be a triangle with $AB < AC$, incenter $I$ and circumcircle $\odot O$. Let $M$ be the midpoint of the arc $BAC$ and $N$ the midpoint of the arc $BC$ not containing $A$. The point $D$ on $\odot O$ is such that $AD \parallel BC$, and $E$ is the point where the $A$-excircle touches $BC$. A point $F$ inside the triangle satisfies $FI \parallel BC$ and $\angle BAF = \angle EAC$. The line $NF$ meets $\odot O$ again at $G$, the line $AG$ meets the line $IF$ at $L$, and the lines $AF$ and $DI$ meet at $K$. Prove that $ML \perp NK$.
The diagram
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