Euclidean and napoleonian theorems: their derivation from pythagorean

This research focuses mainly on the three most elegant statements of triangle geometry, namely, Pythagorean, Euclidean, and Napoleonian theorems. In classical geometry, Pythagorean Theorem states that if one is to construct squares to each of the sides of any right triangle, then the area of the square constructed at the hypotenuse is equal to the sum of the areas of the other two squares constructed on the other two sides, while Euclidean Theorem states that if one is to construct similar figures to each of the sides of any right triangle, then the area of the figure constructed at the hypotenuse is equal to the sum of the areas of the other two figures constructed on the other two sides. On the other hand, Napoleonian Theorem states that if one is to construct equilateral triangles on the sides of any triangle, the centers of those equilateral triangles themselves form an equilateral triangle. Here, the researcher investigates the three above-said theorems and in the process provides proof of Euclidean Theorem using the Pythagorean Theorem, then goes to prove Napoleonian Theorem using again the Pythagorean Theorem.