PlanetMath: proof of parallelogram law

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proof of parallelogram law

(Proof)

The proof follows directly from Apollonius theorem noticing that each diagonal is a median for the triangles in which parallelogram is split by the other diagonal. Also, the diagonals bisect each other.

$\includegraphics{parallelogramlaw}$

Therefore, Apollonius theorem implies

$\begin{displaymath}2\left(\frac{d_1}{2}\right)^2 +\left(\frac{d_2}{2}\right)^2=u^2+v^2.\end{displaymath}$

Multiplying both sides by

and simplification leads to the desired expression.

"proof of parallelogram law" is owned by Mathprof. [ full author list (2) | owner history (2) ]

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Cross-references: expression, sides, implies, parallelogram, triangles, median, diagonal, Apollonius theorem, proof

This is version 2 of proof of parallelogram law, born on 2002-05-30, modified 2006-10-15.
Object id is 2971, canonical name is ProofOfParallelogramLaw.
Accessed 9870 times total.

Classification:

51-00 (Geometry :: General reference works )

Pending Errata and Addenda

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