If the legs of a right triangle have lengths a and b,
and the hypotenuse has length c,
the lengths satisfy a^{2} + b^{2} = c^{2}.
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 a^{2} + b^{2} = c^{2} 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 c |