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 (X, Y, Z, W) of equally spaced points. Let T be the common midpoint of arcs YZ and XW.

For definiteness, assume Y and Z are red. Then, if any X, W, T is also red, a trio of equally spaced red points is determined. Otherwise, (X, T, W) itself is a trio of equally spaced blue points.

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 in the midpoint of the sought after trio.