Question 3
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the segment $AM$. The circumcircles of $BPM$ and $CPM$ meet $\Gamma$ again at $D$ and $E$, respectively. The lines $DP$ and $EP$ meet, a second time, the circumcircles of $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that, as $P$ varies, the circumcircle of $AXY$ passes through a fixed point $T\ne A$.
Editorial clarification: take $P$ in the interior of $AM$ and restrict to positions where the distinct second intersections and the circumcircle $AXY$ are defined. The endpoint wording in the original source is preserved above.
Editorial clarification: take $P$ in the interior of $AM$ and restrict to positions where the distinct second intersections and the circumcircle $AXY$ are defined. The endpoint wording in the original source is preserved above.