KöMaL 2016 B4800

$T$ is a point on line $BC$ , different from the midpoint of line segment $BC$ . Circle $k$ is centred at $T$ , and $A$ is its intersection with the perpendicular drawn to $BC$ at $T$ . The intersections of $k$ with the lines $AB$ and $AC$ are $K$ and $L$ , respectively. Let $k$ intersect the circumscribed circle of $ABC$ again at $M$ . Prove that the lines $KL$ , $AM$ and $BC$ are concurrent.