Proof of the Pythagorean theorem
This great rule of thumb was known to Chinese, Indian, and Mesopotamian mathematicians, and helped build the great pyramids of Egypt. For any right triangle, the square of the hypotenuse is the sum of the squares of the other two sides. But Pythagoras could have been the first person to offer a proof of it, rearranging in a square four copies of the triangle.
A circle’s diameter and the chords from its end points to a common point on the circle construct a right triangle. Ancient trigonometry derived from the chord function. The chord of an angle between two radii times the radius of the circle is the distance between the two points on the circle. The chord of 60° is the radius. The sum of the squares of the chords of supplementary angles (whose sum is 180°) is the square of the diameter.
Infinitely many sets of three whole numbers satisfy the Pythagorean theorem and many more are not even rational.