# proof of Apollonius theorem

Let $b=\overline{CA}$, $a=\overline{BC}$, $c=\overline{AB}$, and $m=\overline{AM}$. Let $\angle CMA=\theta$, so that $\angle BMA=\pi-\theta$.

By the law of cosines, $b^{2}=m^{2}+\frac{a^{2}}{4}-am\cos\theta$ and $c^{2}=m^{2}+\frac{a^{2}}{4}-am\cos(\pi-\theta)=m^{2}+\frac{a^{2}}{4}+am\cos\theta$, and adding gives

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

QED

Title proof of Apollonius theorem ProofOfApolloniusTheorem 2013-03-22 12:41:16 2013-03-22 12:41:16 quincynoodles (312) quincynoodles (312) 4 quincynoodles (312) Proof msc 51-00 ProofOfApolloniusTheorem2 LengthsOfTriangleMedians