Question 16
As described in Fig. 2.1, let $ABC$ be an acute triangle with $AB\lt AC$, incentre $I$, and circumcircle with centre $O$. Let $M,N$ be the midpoints of arc $BAC$ and of arc $BC$ not containing $A$, respectively. Let $D$ lie on arc $AC$ with $AD\parallel BC$. The $A$-excircle touches $BC$ at $E$. A point $F$ inside triangle $ABC$ satisfies $IF\parallel BC$ and $\angle BAF=\angle CAE$. The line $NF$ meets the circumcircle again at $R\ne N$, the lines $AF$ and $DI$ meet at $K$, and the lines $AR$ and $IF$ meet at $L$. Prove that $NK\perp ML$.
Source wording correction: $\odot O$ denotes the circumcircle with centre $O$, not the circumcentre itself. The phrase “inscribed circle ... against $\angle BAC$” is rendered as the $A$-excircle, consistently with the source proof's explicit use of excircle contact lengths $BZ=CE$; it is not the incircle. Figure labels 2.1–2.3 occur in the source text, but the record contains no corresponding image or image link.
Source wording correction: $\odot O$ denotes the circumcircle with centre $O$, not the circumcentre itself. The phrase “inscribed circle ... against $\angle BAC$” is rendered as the $A$-excircle, consistently with the source proof's explicit use of excircle contact lengths $BZ=CE$; it is not the incircle. Figure labels 2.1–2.3 occur in the source text, but the record contains no corresponding image or image link.