HMMT 2020 G9

Circles $\omega_{a}, \omega_{b}, \omega_{c}$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D \in B C, E \in C A, F \in A B$ ). Lines $B E$ and $C F$ meet at $T$. Given that $\omega_{a}$ has radius 341 , there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius 49 tangent to all three circles, compute the distance from $T$ to $\ell$.