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?
1. "RE: Coloring Points in a plane"
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>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.
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