# Parallelogram Law The **parallelogram law** relates the lengths of the diagonals of a parallelogram to the lengths of its sides. +-- {: .image} [[ParallelogramLaw.png:pic]] =-- Specifically, as in the above diagram let $a$ and $b$ be the two side lengths of a parallelogram, and $c$ and $d$ the lengths of its diagonals. Then $$ c^2 + d^2 = 2 a^2 + 2 b^2 $$ ## Proof from the Cosine Rule +-- {: .image} [[ParallelogramLawProof.png:pic]] =-- The parallelogram law can be deduced from the [[cosine rule]] by considering two triangles inside the parallelogram. Since the two angles add up to $180^\circ$, we have $\cos(D) = - \cos(C)$ and so from the cosine rule: $$ \begin{aligned} c^2 &= a^2 + b^2 - 2 a b \cos(C) \\ d^2 &= a^2 + b^2 - 2 a b \cos(D) = a^2 + b^2 + 2 a b \cos(C) \\ c^2 + d^2 = 2 a^2 + 2 b^2 \end{aligned} $$ ## Proof by Similar Triangles +-- {: .image} [[ParallelogramLawProofByTriangles.png:pic]] =-- Using the two triangles formed by cutting the parallelogram along each of its diagonals it is possible to form a new parallelogram. The top edge of this parallelogram has length $c^2 + d^2$ while the bottom edge has length $a^2 + b^2 + a^2 + b^2 = 2 a^2 + 2 b^2$.