The scalene triangle $ABC$ has incenter $I{}$ and circumcenter $O{}$. The points $B_A$ and $C_A$ are the projections of the points $B{}$ and $C{}$ onto the line $AI$. A circle with a diameter $B_AC_A$ intersects the line $BC$ at the points $K_A$ and $L_A$. Prove that the circumcircle of the triangle $AK_AL_A$ touches the incircle of the triangle $ABC$ at some point $T_A$. Define the points $T_B$ and $T_C$ analogously. Prove that the lines $AT_A,BT_B$ and $CT_C$ intersect on the line $OI$.