Question 1
Let $ABC$ be a triangle and $\Gamma$ its circumcircle. Let $\omega_{A}$ be the circle internally tangent to $(AB)$, $(AC)$ and to $\Gamma$. Denote by $T_{A}$ the point of tangency of $\Gamma$ with $\omega_{A}$. Define $T_{B}$ and $T_{C}$ similarly.
Prove that $\left(AT_{A}\right),\left(BT_{B}\right)$ and $\left(CT_{C}\right)$ are concurrent.
Prove that $\left(AT_{A}\right),\left(BT_{B}\right)$ and $\left(CT_{C}\right)$ are concurrent.