# Two Colors - Three Points on Circle

*equally spaced*points of the same color.

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Copyright © 1996-2018 Alexander Bogomolny*equally spaced*points of the same color.

Of any three points two must be of the same color. Therefore, we can definitely find points of the same color as close together as we please. Accordingly, let Y and Z be of the same color and so close that there is room to extend the pair into the ordered quadruple

For definiteness, assume Y and Z are red. Then, if any

Here's a solution by V. Paul Smith, Jr. Consider the vertices of a regular pentagon inscribed in the circle. 3 of the points (at the vertices) have to be the same color. Those 3 points form the vertices of an isosceles triangle, which by definition has two congruent sides. The point common to both matching sides is the midpoint of the sought after trio.

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Copyright © 1996-2018 Alexander Bogomolny72022698