>I was reading your proofs of Coloring Points in the Plane,
>and I'm having trouble understanding the second solution.
>Why is it that if the circle has radius twice that of the
>altitude of an equilateral triangle, any point on that
>circle cannot be the same color as the center? (The proof
>says if the points on the circle are the same color as the
>center, then there's nothing to prove) Why?

We are looking for two points of the same color at a distance smaller than the radius of a circle. Such a circle contains a chord of the given distance. This solves the problem in case all points on the circle are of the same color.