Problem

Source: German TST 2022, probably on exam 6-7, again proposed by me

Tags: geometry, perpendicular bisector, Triangle Geometry, TST



Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$. The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$. The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$. The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$. (Yes, these definitions have the symmetries you would expect.) Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.