KöMaL 2022 B5285

In an acute-angled triangle $ABC$, $AB=AC$. Points $A'$, $B'$ and $C'$ are moving along the circumscribed circle of the triangle, so that triangle $A'B'C'$ always remain congruent to triangle $ABC$, and have the same orientation. Let $P$ be the intersection of lines $BB'$ and $CC'$. Show that the lines $A'P$ all pass through a certain point.