Napoleon’s theorem offers a tour de force for constraint geometry. The theorem states that for any arbitrary triangle, if you construct an equilateral triangle on each edge, and join the centers of the incircles of these triangles, then the resulting triangle is equilateral. The theorem is named for, and supposedly discovered by, Napoleon Bonaparte, himself no stranger to tours de force.
Take a triangle and draw a triangle on each of its sides
We can make the subtended triangles equilateral simply by specifying congruence constraints. Start by making AC congruent to AF and AC congruent to FC… (Congruence is the second option on the Measurement Selection drop down button when two segments are selected.)
We can follow the same procedure for the other two triangles:
We have created the required equilateral triangles. Now for the incircles. We sketch the circles then set tangency constraints - three for each.
Now let’s join the centers of these circles. Shading the resulting triangle makes it stand out from the cat’s cradle of lines and circles. If you’re not convinced that it is indeed equilateral - and why should you be, Napoleon was more famous for geopolitics than geometry - inspect its side lengths
Dragging the original triangle shows that the derived one still remains equilateral, obedient as a Grenadier of the Old Guard to his Emperor.
