$MO \cap (O) = E, F$. $MAB$ and $MN$ are respectively a secant and a tangent of $(O)$ at $N$ ($ME < MA < MN < MB < MF$).The tangent of $(O)$ at $E$ intersects semicircle diameter $MF$ at $K$. $NO \cap FK = T$. Prove that the circumcentre of $\triangle ABT$, $\triangle EFT$ and the midpoint of $KT$ collinear.
I am trying to prove the following claims leading to the solution of the problem.
$KN \perp MT$
$MT \perp PQ$ (or $MT$ is the tangent of two externally circumcircles of $\triangle ABT$ and $\triangle EFT$)
$RS \parallel KN$ (or $S$ is the midpoint of $NT$ with $S = NT \cap PQ$, $P$ and $Q$ are respectively the circumcentre of $\triangle ABT$ and $\triangle EFT$)
This is supposed to be an easy problem. Why couldn't I solve it?