Question 1
Let $\gamma_1$ and $\gamma_2$ be two circles tangent at point $T$, and let $\ell_1$ and $\ell_2$ be two lines through $T$. The lines $\ell_1$ and $\ell_2$ meet again $\gamma_1$ at points $A$ and $B$, respectively, and $\gamma_2$ at points $A_1$ and $B_1$, respectively. Let further $X$ be a point in the complement of $\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2$. The circles $ATX$ and $BTX$ meet again $\gamma_2$ at points $A_2$ and $B_2$, respectively. Prove that the lines $TX, A_1B_2$ and $A_2B_1$ are concurrent.