BALTIC-WAY 2025 Problem 15

In an acute, scalene triangle $ABC$, the perpendicular bisector of $BC$ intersects $AC$ and $AB$ in $A_b$ and $A_c$, respectively. Define $B_c, B_a, C_a$ and $C_b$ similarly. Prove that the circumcentres of $ABC, A_bB_cC_a$, and $A_cB_aC_b$ are collinear.