Finding a monochromatic right angled triangle of any side.

Question

If every point in the plane R^2 is colored red or blue.Prove that given a^2+b^2=c^2 a,b,c are real numbers, one can find a monochromatic triangle with the sides a,b,c respectively. (Also is the answer true for more than 2 colors. Provide a proof or counter example adequately.)