alternate proof of parallelogram law

Proof of this is simple, given the cosine law:


where a, b, and c are the lengths of the sides of the triangle, and angle ϕ is the corner angle opposite the side of length c.

Let us define the largest interior anglesMathworldPlanetmath as angle θ. Applying this to the parallelogramMathworldPlanetmath, we find that

d12 = u2+v2-2uvcosθ
d22 = u2+v2-2uvcos(π-θ)

Knowing that


we can add the two expressions together, and find ourselves with

d12+d22 = 2u2+2v2-2uvcosθ+2uvcosθ
d12+d22 = 2u2+2v2

which is the theorem we set out to prove.

Title alternate proof of parallelogram law
Canonical name AlternateProofOfParallelogramLaw
Date of creation 2013-03-22 12:43:52
Last modified on 2013-03-22 12:43:52
Owner drini (3)
Last modified by drini (3)
Numerical id 5
Author drini (3)
Entry type Proof
Classification msc 51-00
Related topic ProofOfParallelogramLaw2