<Tri-2025-2273>, hint

Let O', O" be the circumcenters of triangles ABB', ACC' and M, N be the midpoints of AB, AC, respectively.

Since  AB=AB' O' lies on AC , an intersection with perpendicular at M which passes through

O, the circumcenter of ABC. Similarly O" lies on AC, an intersection with  perpendicular 

at N. Hence O lis on O'O", which passes through the midpoint of PA, that is O.