In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the...

a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are...

of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q...

p^{2}+q^{2}=2(m^{2}+n^{2}).} This is also a corollary to the parallelogram law applied in the Varignon parallelogram. The lengths of the bimedians can also be expressed...

product. The norm associated with any inner product space satisfies the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . {\displaystyle...

rest d0 ones are degenerate and form the U ∩ U⊥ intersection. The parallelogram law takes the form q ( x ) + q ( y ) = 1 2 ( q ( x + y ) + q ( x − y )...

Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection...

theorem in an inner product space to non-orthogonal vectors is the parallelogram law: 2 ‖ v ‖ 2 + 2 ‖ w ‖ 2 = ‖ v + w ‖ 2 + ‖ v − w ‖ 2   , {\displaystyle...

of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was...

{\displaystyle x} and y {\displaystyle y} are linearly dependent. Parallelogram law ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 {\displaystyle...

the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law. The theorem can be proved as a special...

convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem...

distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional...

lengths of the tangent vectors fulfil the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the...

+i\left(\|x+iy\|^{2}-\|x-iy\|^{2}\right)\right).} To see that the parallelogram law is sufficient, one observes in the real case that ⟨ x , y ⟩ {\displaystyle...

commutant) Commutative diagram Commutative (neurophysiology) Commutator Parallelogram law Particle statistics (for commutativity in physics) Proof that Peano's...

norm does not arise from any inner product due to not satisfying the parallelogram law.) It is not entirely straightforward to construct a Hilbert space...

convexity of F. Same for strict convexity. Linearity in F, law of cosines, parallelogram law: by definition. Duality: See figure 1 of. Bregman balls are...

of a triangle Euler's quadrilateral theorem, an extension of the parallelogram law to convex quadrilaterals Euler–Rodrigues formula concerning Euler–Rodrigues...

British flag theorem for rectangles equates two sums of two squares The parallelogram law equates the sum of the squares of the four sides to the sum of the...

generally, a polytope (in arbitrary finite dimension). The parallelogram law (called also parallelogram identity) ‖ x − y ‖ 2 + ‖ x + y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖...

there to exist such an inner product is for the norm to satisfy the parallelogram law: ‖ x + y ‖ 2   +   ‖ x − y ‖ 2   =   2 ‖ x ‖ 2 + 2 ‖ y ‖ 2  for all...

tessellation Voronoi tessellation Delaunay triangulation Quasicrystal Parallelogram law Polytope Schläfli symbol Regular polytope Regular Polytopes Sphere...

{\displaystyle f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!} (quadratic equation or parallelogram law) f ( ( x + y ) / 2 ) = ( f ( x ) + f ( y ) ) / 2 {\displaystyle...

p-norm is not an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of p is called Minkowski's...

The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object. When more than two forces are involved...

and c, respectively; if γ is acute, then ab cos γ is the area of the parallelogram with sides a and b forming an angle of γ′ = ⁠π/2⁠ − γ; if γ is obtuse...

on X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} satisfies the parallelogram law, which means that the polarization identity can be used to define...

since any norm that is induced by an inner product should satisfy the parallelogram law ‖ x + y ‖ p 2 + ‖ x − y ‖ p 2 = 2 ‖ x ‖ p 2 + 2 ‖ y ‖ p 2 . {\displaystyle...

through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out...