Question 146
Line $l$ is perpendicular to the side $AC$ of an acute triangle $ABC$ and intersects $AC$ at $K$. Line $l$ intersects the circumscribed circle of $\triangle ABC$ at $P$ and $T$ (point $P$ lies in the same half-plane with respect to $AC$ as vertex $B$). Denote by $P_1$ and $T_1$ the projections of points $P$ and $T$ onto line $AB$. Furthermore, vertices $A, B$ belong to the segment $P_1T_1$. Prove that the center of the circumscribed circle of $\triangle P_1KT_1$ lies on the line containing the midsegment of $\triangle ABC$ parallel to side $AC$.
(Anton Tryhub)
Fig. 31
(Anton Tryhub)
Fig. 31