2025 Putnam B1

Suppose that each point in the plane is colored either red or green, subject to the following condition: For every three noncollinear points $A,B,C$ of the same color, the center of the circle passing through $A,B$ and $C$ is also this color. Prove that all points of the plane are the same color.

假设平面中的每个点都被着色为红色或绿色，且满足以下条件：对于每三个相同颜色的非共线点 $A,B$,C$，穿过$A,B$和$C$ 的圆心也是该颜色。证明平面上所有点的颜色相同。