Prove there's a monochromatic isosceles triangle. The points in a circle are coloured red and blue. Prove that there exists a monochromatic isoceles triangle. I can prove that there exists a monochromatic triangle. If there are no three points of the same colour, then all the remaining points are of the other colour. I have tried to select a point and then draw a line passing through the centre of the circle and the selected point. Now the circle is divided into two semicircles. I call two points mirrors if they lie on different semi circles and have the same distance from the selected point. I have tried to arrive at a contradiction in assuming that there are no mirror points with the same colour as the selected point but it's not successful.

Inscribe a regular pentagon in the circle. Three of the vertices must have the same color and they form the triangle you seek.