59th Ukrainian National Mathematical Olympiad

Line $l$ is perpendicular to the side $AC$ of acute triangle $ABC$, and intersects $AC$ at $K$. $l$ intersects circumscribed circle of $△ABC$ at $P$ and $T$ (point $P$ in the same half-plane w.r.t. $AC$ as vertex $B$). By $P_1$ and $T_1$ are denoted projections of points $P$ and $T$ to line $AB$. Furthermore, vertices $A,B$ belong to the segment $P_1{T}_1$. Prove that the center of circumscribed circle of $△P_{1}K{T}_1$ lies on the line, which contains midsegment of $△ABC$, parallel to the side $AC$. (Anton Tryhub)

Fig. 31