In
mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary
geometry. It states that the sum of the squares of the lengths of the four sides of a
parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: ''AB'', ''BC'', ''CD'', ''DA''. But since in
Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, ''AB'' = ''CD'' and ''BC'' = ''DA'', the law can be stated as
If the parallelogram is a
rectangle, the two diagonals are of equal lengths ''AC'' = ''BD'', so
and the statement reduces to the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
. For the general
quadrilateral with four sides not necessarily equal,
where
is the length of the
line segment joining the
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
s of the diagonals. It can be seen from the diagram that
for a parallelogram, and so the general formula simplifies to the parallelogram law.
Proof
In the parallelogram on the right, let AD = BC = ''a'', AB = DC = ''b'',
By using the
law of cosines in triangle
we get:
In a parallelogram,
adjacent angles are
supplementary, therefore
Using the
law of cosines in triangle
produces:
By applying the
trigonometric identity to the former result proves:
Now the sum of squares
can be expressed as:
Simplifying this expression, it becomes:
The parallelogram law in inner product spaces
In a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, the statement of the parallelogram law is an equation relating
norms:
The parallelogram law is equivalent to the seemingly weaker statement:
because the reverse inequality can be obtained from it by substituting
for
and
for
and then simplifying. With the same proof, the parallelogram law is also equivalent to:
In an
inner product space, the norm is determined using the
inner product:
As a consequence of this definition, in an inner product space the parallelogram law is an algebraic identity, readily established using the properties of the inner product:
Adding these two expressions:
as required.
If
is orthogonal to
meaning
and the above equation for the norm of a sum becomes:
which is
Pythagoras' theorem.
Normed vector spaces satisfying the parallelogram law
Most
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
and
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm for a vector
in the
real coordinate space is the
-norm:
Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the
-norm if and only if
the so-called norm or norm.
For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product the ...
. In the real case, the polarization identity is given by:
or equivalently by
In the complex case it is given by:
For example, using the
-norm with
and real vectors
and
the evaluation of the inner product proceeds as follows:
which is the standard
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
of two vectors.
Another necessary and sufficient condition for there to exist an inner product that induces the given norm
is for the norm to satisfy
Ptolemy's inequality
In Euclidean geometry, Ptolemy's inequality relates the six distances determined by four points in the plane or in a higher-dimensional space. It states that, for any four points , , , and , the following inequality holds:
:\overline\cdot \overli ...
:
See also
*
*
*
*
*
*
*
References
External links
*
The Parallelogram Law Proven Simplya
Dreamshire blogThe Parallelogram Law: A Proof Without Wordsat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
{{DEFAULTSORT:ParallelogramLaw
Euclidean geometry
Theorems about quadrilaterals