Generalization 1: The three equilateral triangles don't need to be of equal size as shown below. In other words, the outer vertices don't have to be cyclic.
Generalization 2: The result can even be further generalized to four attached equilateral triangles as shown below.
Here is a nice animated dynamic sketch displaying the second generalization.
Challenge
1) Can you prove these generalizations purely geometrically?
2) Can you think of other possible generalizations or variations? Explore!
Hint: Try using spiral similarities (a combination of rotations and dilations) to prove the above.
Here's a simple variation of Generalization 2 above using regular hexagons instead of equilateral triangles ...
Comment: The first generalization above has been called the Asymmetric Propellor theorem by Martin Gardner (1999) and can be seen to be equivalent to Napoleon's Theorem. Apparently the result is originally due to the nineteenth-century British mathematicians William Kingdom Clifford and Arthur Cayley. In 1973 Leon Bankoff, Paul Erdös and Murray Klamkin generalized the result further to the second generalization, as well as some further generalizations involving similar triangles. For more information go to: Similarity.