## Napoleon's triangle

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Napoleon's triangle is found by constructing equilateral triangles on the sides of a triangle and then connecting the centers of the equilateral triangles. Since all the centers, centroid, orthocenter, incenter, and circumcenter, are equal in an equilateral triangle, it is only necessary to find one of the centers. The result of connecting the centers is Napoleon's triangle. The following are investigations of Napoleon's triangle done in GSP.

Construction of Napoleon's triangle Napoleon's triangle is the yellow triangle. This yellow triangle is equilateral. Whenever the centers of equilateral triangles are connected, we get an equilateral triangle.

Investigations of Napoleon's Triangle

Let's see what happens to Napoleon's triangle for various original triangles. First, let's examine what happens when the original triangle is equilateral. From observing the above diagram, one notices that Napoleon's triangle is congruent to the original triangle by the AAA postulate. Similarly, Napoleon's triangle is congruent to the exterior triangles. Now, let's see what happens when the original triangle is isosceles. Again, Napoleon's triangle, the red triangle, is equilateral.
What happens when the original triangle is a right triangle? Again, Napoleon's triangle, the yellow triangle, is equilateral.

In all of the cases that I have explored, Napoleon's triangle was always equilateral.