Get some paper and pen and start coloring. Try to avoid a monochromatic triangle, and see that you are forced to make one anyway.
Every triangle must have two of one color and one of the other. Let's choose the largest triangle, as it seems to have the most intersection with the other triangles. As there's nothing special about either color, let there be two blue and one red:
The top most little triangle now has 1 red, so the other two must either be both blue, or one red one blue.
Now we have 3 triangles which have two blue circles, which force their third circle to be red:
Oh no! We have been forced to make a red triangle.
We have one red and one blue in our upper triangle. By symmetry it doesn't matter which is which.
And again we color in what we're forced to to avoid a monochromatic triangle.
And again we have been forced to make a triangle with all the same color. Thus, no matter how we proceed with the coloring, there is a monochromatic triangle.