Each of the vertices of a regular nonagon has been colored either red or blue. Prove that there exist two congruent monochromatic triangles.
We call a monochromatic triangle red (blue) if all of its vertices are red (blue). Because there are nine vertices colored in two colors, at least five must be of the same color. Without loss of generality, we say that this color is red. Hence there are at least $\binom52=10$ red triangles. Now, I have to prove that among these $10$ triangles, there are two congruent triangles, which I can't imagine how am I supposed to prove. Please help.
N.B.: Solution containing application of bijection principle will be much appreciated. Thank you.