Napoleon’s theorem is usually presented as I presented it in the previous post.

You start with a triangle (solid blue) and add equilateral triangles (dashed green) on the **outside** of the triangle. When you connect the centroids of these triangles you get a (dotted red) equilateral triangle.

But Napoleon’s theorem is more general than this. It says you could also add the triangles to the **inside**. The result is much harder to parse visually. The following diagram flips each green triangle over.

You still get an equilateral triangle when you connect the centroids, but it’s a different triangle.

Can rhis be generalized to isosceles triangles with the same top angle? If so, which of the three centers?

what happens in between … interpolating from outside to inside? might have to try geogebra to viz this …