RO-TST 2010 Problem 8

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.