NZ-NZMO 2025 Problem 7

Let $ABC$ be a triangle and let $D$ be a point inside the triangle $ABC$ such that $AD$ bisects $\angle BAC$. Let line $BD$ meet side $AC$ at $E$. Let line $CD$ meet side $AB$ at $F$. Let $T$ be the intersection of the (internal) angle bisectors of $\angle AED$ and $\angle AFD$. Prove that if $T$ lies on segment $AD$, then triangle $ABC$ is isosceles.