KöMaL 2022 B5282

In an acute-angled triangle $ABC$, the foot of the altitude drawn from vertex $A$ is $T$. The projection of $T$ on side $AB$ is $D$, and its projection on side $AC$ is $E$. Let $F$ be the intersection of side $BC$ and the circle $ABE$ that is different from $B$. Analogously, let $G$ be the intersection of side $BC$ and circle $ACD$, different from $C$. Show that $TF=TG$.