Problem

Source: Argentina Cono Sur TST 2024 P3

Tags: geometry, circumcircle



Let $ABC$ be an acute triangle. The point $B'$ of the line $CA$ is such that $A$, $C$, $B'$ are in that order on the line and $B'C=AB$; the point $C'$ of the line $AB$ is such that $A$, $B$, $C'$ are in that order on the line and $C'B=AC$. Show that the circumcenter of triangle $AB'C'$ belongs to the circumcircle of triangle $ABC$.