
It is believed
that this theorem was known, in some form, long before the
time of Pythagoras. A thousand years before
Pythagoras, the Babylonians recorded their knowledge of the
theorem on clay tablets. The Egyptians used the
concept of the theorem in the building of the pyramids.
In 1100 B.C. China, TschouGun knew of this theorem.

It was, however,
Pythagoras who generalized the theorem to all right triangles
and is credited with its first geometrical demonstration.
Consequently, the theorem bears his name.

Geometrical Proof:
The Pythagorean Theorem has drawn a good deal of attention
from mathematicians. There are hundreds of geometrical
proofs (or demonstrations) of the theorem, with even a
larger number of algebraic proofs.
Geometrically,
the Pythagorean Theorem can be interpreted as discussing the
areas of squares whose sides are the sides of the triangle
(as seen in the picture at the left). The theorem can
be rephrased as, "The (area of the) square described upon
the hypotenuse of a right triangle is equal to the sum of
the (areas of )the squares described upon the other two
sides." 

Converse:
Theorem: If a triangle
is a right triangle, the square of the length of the hypotenuse is
equal to the sum of the squares of the lengths of the legs.
Converse: If the square
of the length of the longest side of a triangle is equal to the sum
of the squares of the lengths of the other two sides, the triangle
is a right triangle.
Pythagorean Triples: There
are certain sets of numbers that have a very special property in
connection to the Pythagorean Theorem.
Not only do these numbers satisfy the Pythagorean Theorem, but
any multiples of these numbers also satisfy the Pythagorean
Theorem.
For example:
the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you
multiply all three numbers by 2 (you will get 6, 8, and 10), these new numbers
ALSO satisfy the Pythagorean theorem.
The special sets of numbers that
possess this property are called
Pythagorean Triples.
The most common Pythagorean
Triples are:
3, 4, 5 
5, 12, 13 
8, 15, 17 
There are equations for
generating
Pythagorean Triples. Such equationss are used by
computer programmers to generate lists of Pythagorean Triples.
Equations: When m and n are both positive integers and
m < n, Pythagorean Triples can be generated using
the following equations where a and b will be the legs
of the right triangle and c will be the hypotenuse.
It can be shown that the results from
these equations do, in fact, satisfy the Pythatorean Theorem:
Connection to the
Distance Formula: The
Pythagorean Theorem serves as the basis for determining the distance
between two points in the Cartesian plane, called the
Distance Formula.

If (x_{1},
y_{1}) and (x_{2},
y_{2}) are points in the plane,
then the distance between them can be determined
using the Pythagorean Theorem.
