Threesomes

Perhaps the best-known geometric result ever established is one that dates back to antiquity. More than 2,500 years ago, the Greek mathematician Pythagoras verified that if triangle ABC has a right angle at C, then a + b² = c², where a, b, and c are the lengths of the sides of the triangle that are opposite vertices A, B, and C, respectively. This statement, called the Pythagorean Theorem in honor of its discoverer, has been reconfirmed in many different ways throughout history. One particularly nice proof is a direct consequence of the following sketch: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315223612/789951b4-8233-44bb-8b22-44ba80370387/content/ufig13.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Observing that the area of the large square must equal the sum of the areas of the small square and the four right triangles, we find that https://www.w3.org/1998/Math/MathML"> ( a+b ) 2 = c 2 + 4 ( 1 2 ab ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315223612/789951b4-8233-44bb-8b22-44ba80370387/content/eq1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Upon simplification, this yields the desired result.