KöMaL 2021 C1666

In an acute-angled triangle $ABC$, let $K$ and $D$, respectively, be the intersections of the interior angle bisector drawn from point $A$ with the interior angle bisector drawn from $B$, and with side $BC$. The perpendicular drawn to angle bisector $AD$ at point $K$ intersects side $AB$ at point $E$. $F$ is the foot of the perpendicular drawn from point $E$ to $BC$. $T$ is the foot of the perpendicular drawn from point $D$ to line $AB$. Prove that $T$ lies on the circumscribed circle of triangle $KEF$.