The document summarizes a proof of the Pythagorean theorem. It states that by placing four identical right triangles together to form the sides of a larger square, the total area of the triangles is shown to equal the area of the larger square. This allows the area of the larger square, which is the square of the hypotenuse c, to be expressed as the sum of the squares of the legs a and b, proving the Pythagorean theorem that a2 + b2 = c2.

1. The Pythagorean theorem states that the squares of both legs of a right triangle add
up to the square of the hypotenuse, or more famously, a2 + b2 = c2. We can take
advantage of the fact that c2 equals the area of a square with side length c formed by
a hypotenuse of length c. If we can express the area of the square c2 in terms of a and
b, the leg lengths of the right triangle with hypotenuse length c, we can prove the
Pythagorean theorem.
Because a square has four sides, we put four identical right
triangles in the square so that their hypotenuses form the
square’s sides. If the triangle has 45-degree angles and identical
legs, no smaller square will be formed in the center of the
larger square with side length c. Right triangles with any other
angles will form a square in the middle of the larger square
because diagonals can only be formed with 45 degree angles.
Because the angles opposite a and b add up to 90 degrees, side
b forms part of a, assuming a is the longer of the two legs. The
area of each triangle is ab/2. Multiplied by 4, the total area of
the triangles is 2ab. The side length of the square formed in the
middle is a - b, since b forms part of a. Thus, the area of the
small square is (a – b)2 and the area of the large square in terms
of a and b is 2ab + (a – b)2 . (a - b)2 simplifies to a2 - 2ab + b2 and
the area of the large square in terms of a and b simplifies to a2 +
b2.
Area of large square = c2 = a2 + b2 , where c is the hypotenuse of the triangle and a
and b are its legs.
*Picture taken from http://www.cut-the-knot.org/pythagoras/index.shtml