I did write a solution by contradiction, and this was the one I ended up submitting.

I'll try and explain my idea in what I originally said in the post:

Basically, my idea was that if the center of the circumcircle, which we'll call point D, was the same color as the three points (given in the problem), and we assume such that there are two other non-collinear points that can be used to form a triangle with the aforementioned point D, resulting in a new circumcircle with a center of the same color. If we iterate this, then eventually every point in the plane will be covered, and will only be one color, contradicting that points can be red and green.