# proof of Apollonius theorem

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

By Stewart’s theorem we have

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

and thus

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

Multiplying both sides by $2$ gives

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

QED

 Title proof of Apollonius theorem Canonical name ProofOfApolloniusTheorem1 Date of creation 2013-03-22 12:41:20 Last modified on 2013-03-22 12:41:20 Owner drini (3) Last modified by drini (3) Numerical id 5 Author drini (3) Entry type Proof Classification msc 51-00 Related topic Median Related topic Cevian Related topic ApolloniusTheorem Related topic StewartsTheorem Related topic ProofOfStewartsTheorem Related topic ProofOfApolloniusTheorem