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:

formula for the length of the median

Reference

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

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