The parallelogram law relates the lengths of the diagonals of a parallelogram to the lengths of its sides.
ParallelogramLaw.png
Specifically, as in the above diagram let and be the two side lengths of a parallelogram, and and the lengths of its diagonals. Then
ParallelogramLawProof.png
The parallelogram law can be deduced from the cosine rule by considering two triangles inside the parallelogram. Since the two angles add up to , we have and so from the cosine rule:
ParallelogramLawProofByTriangles.png
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 while the bottom edge has length .