# Length of a Median

Given triangle $ABC$, with sides $a,b,c$ opposite vertices $A,B,C.$ The length of the median $m_a$ from $A$ is given by

$\displaystyle m_{a}^{2}=\frac{b^{2}+c^{2}}{2}-\frac{a^2}{4},$ or $\displaystyle m_{a}=\frac{1}{2}\sqrt{2(b^{2}+c^{2})-{a^2}}.$

Here is a proof based on the Parallelogram Law:

### Reference

1. C. P. Lawes, Proof Without Words: The Length of a Triangle Median via the Parallelogram Law, Math. Mag. 86 (2013), 146