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