PlanetMath: proof of Apollonius theorem

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proof of Apollonius theorem

(Proof)

Let be a median of the triangle, as shown in the figure.

$\includegraphics{apollonius}$

$\begin{displaymath}a\left(m^2+ \left(\frac{a}{2}\right)^2 \right)=b^2\left(\frac{a}{2}\right)+c^2\left(\frac{a}{2}\right)\end{displaymath}$

and thus

$\begin{displaymath}m^2+\left(\frac{a}{2}\right)^2=\frac{b^2+c^2}{2}.\end{displaymath}$

Multiplying both sides by gives

$\begin{displaymath}2m^2+\frac{a^2}{2}=b^2+c^2.\end{displaymath}$

"proof of Apollonius theorem" is owned by drini. [ owner history (1) ]

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Cross-references: QED, sides, Stewart's theorem, triangle, median

This is version 2 of proof of Apollonius theorem, born on 2002-05-29, modified 2002-12-27.
Object id is 2968, canonical name is ProofOfApolloniusTheorem2.
Accessed 7938 times total.

Classification:

51-00 (Geometry :: General reference works )

Pending Errata and Addenda

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