Pythagorean theorem

The Pythagorean theorem or Pythagoras' theorem is named after and commonly
attributed to the 6th century BC Greek philosopher and mathematician
Pythagoras, though the facts of the theorem were known before he lived. The
theorem states:

     The sum of the areas of the squares on the legs of a right
     triangle is equal to the area of the square on the hypotenuse.

(A right triangle is one with a right angle; the legs are the two sides that
make up the right angle; the hypotenuse is the third side opposite the right
angle; the square on a side of the triangle is a square, one of whose sides
is that side of the triangle).

Since the area of a square is the square of the length of a side, we can
also formulate the theorem as:

     Given a right triangle, with legs of lengths a and b and
     hypotenuse of length c, then

          a2 + b2 = c2

Draw a right triangle with sides a, b, and c. Then take a copy of this triangle and 
place its a side in line with the b side of the first, so that their c sides form a right
angle (this is possible because the angles in any triangle add up to two right angles --
think it through). Then place the a side of a third triangle in line with
the b side of the second, again in such a manner that the c sides form a
right angle. Finally, complete a square of side (a+b) by placing the a side
of a fourth triangle in line with the b side of the third. On the one hand,
the area of this square is (a+b)2 because (a+b) is the length of its sides.
On the other hand, the square is made up of four equal triangles each having
area ab/2 plus one square in the middle of side length c. So the total area
of the square can also be written as 4 á ab/2 + c2. We may set those two
expressions equal to each other and simplify:

     [(a+b)^2=4 \cdot ab/2 + c^2]

     a2 + 2ab + b2 = 2ab + c2

     a2 + b2 = c2

Q.E.D.

Note that this proof does not work in non-Euclidean geometries, since, say,
on a sphere, the angles of a triangle don't add up to 180 degrees, and the
above "square" cannot be formed. (See the external links below for a
sampling of the many different proofs of the Pythagorean theorem.)

The converse of the Pythagorean theorem is also true:

     For any three positive numbers a, b, and c such that a2+b2=c2,
     there exists a triangle with sides a, b and c, and every such
     triangle has a right angle between the sides of lengths a and b.

This can be proven using the law of cosines which is a generalization of the
Pythagorean theorem applying to all (Euclidean) triangles, not just
right-angled ones.

Another generalization of the Pythagorean theorem was already given by
Euclid in his Elements:

     If one erects similar figures (see geometry) on the sides of a
     right triangle, then the sum of the areas of the two smaller ones
     equals the area of the larger one.

The Pythagorean theorem stated in Cartesian coordinates is the formula for
the distance between points in the plane -- if (a, b) and (c, d) are points
in the plane, then the distance between them is given by

     [ \sqrt{(a-c)^2 + (b-d)^2} ]

This distance formula generalises to inner product spaces, and the version
of the Pythagorean Theorem in inner product spaces is known as Parseval's
identity.

The Pythagorean theorem also generalizes to higher-dimensional simplexes. If
a tetrahedron has a right angle corner (a corner of a cube), then the square
of the area of the face opposite the right angle corner is the sum of the
squares of the areas of the other three faces. This is called de Gua's
theorem.

Since the Pythagorean theorem is derived from the axioms of Euclidean
geometry, and physical space may not always be Euclidean, it need not be
true of triangles in physical space. One of the first mathematicians to
realize this was Carl Friedrich Gauss, who then carefully measured out large
right triangles as part of his geographical surveys in order to check the
theorem. He found no counterexamples to the theorem within his measurement
precision. The theory of general relativity holds that matter and energy
cause space to be non-Euclidean and the theorem does therefore not strictly
apply in the presense of matter or energy. However, the deviation from
Euclidean space is small except near strong gravitational sources such as
black holes. Whether the theorem is violated over large cosmological scales
is an open problem of cosmology.

Given two vectors, A and B, the Pythagorean Theorem states

     ||A + B||2 = ||A||2 + ||B||2 if and only if the two vectors are
     orthogonal.