Let $ABC$ be an acute triangle. Circle $\omega_1$, with diameter $AC$, intersects side $BC$ at $F$ (other than $C$). Circle $\omega_2$, with diameter $BC$, intersects side $AC$ at $E$ (other than $C$). Ray $AF$ intersects $\omega_2$ at $K$ and $M$ with $AK < AM$. Ray $BE$ intersects $\omega_1$ at $L$ and $N$ with $BL < BN$. Prove that lines $AB$, $ML$, $NK$ are concurrent
Claim : $K,M,L,N$ is cyclic
Proof : Let $NM\cap KL=H$ . Note that $H$ will be the orthocenter of $ABC$ .
By POP, $NH\cdot HM= CH\cdot CF=KH\cdot HL$.
Claim: $C$ is the centre of $(KMLN)$
Proof: Since $CA$ is the diametre , we have CA as the perpendicular bisector of $LN$ .
Similarly $CB$ is the perpendicular bisector of $KM$ .
Now , I just want to show AB is the Polar of $H$ wrt $(KLMN)$ . Then by Brocard's theorem, I know that $NK\cap LM \in AB $.