Team Selection Test for IMO 2023

Let the lines $AT$ and $BT$ intersect the line $CD$ at the points $A_1$ and $B_1$, and intersect the circle $(TCD)$ at the points $A_2$ and $B_2$ (beside $T$). Now one has $\overline{A_{2}D}=\overline{C{B}_2}$ on the circle $(TCD)$, which implies $A_2{B}_2\parallel CD$. On the other hand $A_{1}T\cdot A_1{A}_2=A_{1}C\cdot A_{1}D=A_{1}A\cdot A_{1}K$, hence one gets $A_{1}K/A_1{A}_2=A_{1}T/A_{1}A$ and one analogously gets $B_{1}L/B_1{B}_2=B_{1}T/B_{1}B$. Now $A_{1}T/A_{1}A=B_{1}T/B_{1}B$, therefore $A_{1}K/A_1{A}_2=B_{1}L/B_1{B}_2$, which implies $KL\parallel A_2{B}_2$.