Is there a way I can color this pattern on my computer?

I found this pattern that I wanted to be able to reproduce, for no particular reason other than that I like it and because I thought it would be a good exercise for the Geometer’s Sketchpad program.

I followed the instructions and could reproduce the pattern, provided that I did not try to color it in. Creating the initial pattern takes a bit of work, but is fairly straightforward. Tiling is relatively easy. I combined two triangular patterns to create a parallelogram, and then it is just a mater of tiling parallelograms. The problem, as you can see, came when I tried to color the pattern.

I looked at another geometry drawing program, Geogebra, and it treated colors the same way. To handle overlapping regions, both programs work with layers and opacity. I can give separate colors to the circular regions and the triangle, but I can’t keep them from blending. Is there some program that just allows you to replace the colors in a region by an overlapping region, without doing any blending? It seems like such an obvious way of doing things.