Plane Geometry, The Pythagorean Theorem

The Pythagorean Theorem

If the legs of a right triangle have lengths a and b, and the hypotenuse has length c, the lengths satisfy a2 + b2 = c2.  For example, set a = 33 and b = 56, and the hypotenuse has length 65.  This is the pythagorean theorem.

If the lengths of a right triangle are rational, multiply through by the common denominator and the lengths become integers.  Three integers that satisfy a2 + b2 = c2 are called a pythagorean triple.  All such triples have been characterized.  The simplest triple is a=3 b=4 c=5, also known as the 3 4 5 triangle.  Another example is 12 5 13.

 Given a right triangle with sides a b and c, draw a square a+b units on a side.  Place a copy of the right triangle in each of the four corners of the square.  Each triangle points to the next one, like a snake chasing its tail. Now the bottom of the square, a+b in length, is covered by the a leg of one triangle and the b leg of the next, and similarly for the other three sides.  The region enclosed by the four triangles is a square, c units on a side.  This inner square is tilted relative to the outer square, but it is still a square, having area c2.  The four triangles have a combined area of 2ab, and the outer square has area (a+b)2.  Put this all together and derive a2 + b2 = c2.