by: Doris Santarone
Tesselating Napolean's Triangle
Since the centers of the outwardly constructed equilateral triangles on triangle ABC form an equilateral triangle, then a rotation of 120 degrees will form a tesselation. Below, I started with the white triangle, labeled "original", along with the equilateral triangles on each side of this triangle (and their centers). I used these centers as the center of rotation and continued this process to form the tesselation that you see below. Click here for the GSP Sketch. Move the original triangle's vertices around to see how the tesselation changes, but remains a tesselation.
Focus on one point in the tesselation, say point A labeled below. In order for the original figure to tesselate, the angles that meet at point A (and every other point) must sum to 360 degrees! This is the key to a tesselation!
We know that angles 2, 4, and 6 are 60 degrees from the construction of equilateral triangles. We also know that 
because they are the three angles of the original triangle (see the reference triangle in lower right corner). This means that 
.