Right angles on the incircle chord
Problem
In triangle $ABC$ with incenter $I$, the incircle touches $BC$, $CA$, $AB$ at $D$, $E$, $F$. Let $M$ and $N$ be the midpoints of $BC$ and $AC$, and let the ray $BI$ meet the line $EF$ at $K$. Prove that $BK \perp CK$, and that $K$ lies on the line $MN$.
The diagram
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