These are a few special circumstances that I found while moving point C around. Note that the dotted line segments of the equilateral triangles have been hidden.

This first case is when the original 1st Napolean Triangle has become the 2nd Napolean Triangle.

This shows when the three points of the 2nd Napolean triangle converge to one point. Notice that triangles ABC and GHI appear to be congruent. They are in fact congruent and the single point made from N, M, and O, happens to be the circumcenter, centroid, incenter, and orthocenter for both triangles.

This shows when point C is on top of B. The same holds for when C is on A.

This shows when C is at point M. We have formed 4 congruent triangles with the two Napolean Triangles. Triangles GNO, IOM, HMN, and NMO are all congruent. Also, triangle AOC is congruent to triangle BNC.

This shows when point C is at the midpoint of segment AB. It appears, and is in fact true, that in this case, the two Napolean triangles are congruent.

There are many more interesting findings, try to find as many as you can.

Return