To construct this:
1.) Create any triangle ABC.
2.) Construct equilateral triangles with each side of the original triangle. (I did this by creating a circle around two vertices of the triangle at a time.) For example, the red equilateral triangle was formed by constructing a circle with center A and radius AB. Then a circle with center B and radius AB was constructed. The intersection of the two circles on the exterior of the triangle creates and equilateral triangle sincce all sides are the same radius.
note: I could have also picked the other intersection point of the two circles
3.) This method was used to create equilateral triangles on all 3 sides.
When we clean up the graph and take away some of the construction circles, we get the following picture:
I created this by finding the circumcenter of each triangle and connecting them. Triangle GHI is Napoleon's Triangle.
So, Napoleon's Theorem states that with any triangle, if you construct three equilateral triangles on the sides and connect the centers, you will always form an equilateral triangle. Yes, Napoleon's triangle is EQUILATERAL!
How do we prove this? Let us start with an equilateral triangle.
Repeating this process for the other two sides will yield the same equation, hence, IG = GH = IH! Thus, triangle IGH is equilateral.