3060.54 – Red and Blue Points, Right Angle

Suppose every point in a plane is colored either red or blue. Show that there are three points in the plane which are the same color and which determine a right angle.


If all of the points in the plane are red, then we're done. How many blue points would it take to make it impossible for there to be a red right angle? There must be at least two (and undoubtedly many more, but we just need two at the moment.) In the diagram, A and B are blue. Add points C and D so that lines AC and BD are parallel and are arranged as shown. Name a point E somewhere on either line. No matter whether E is red or blue we have our right angle.