Median Formula

Theorem
The length of a median is equal to

$\displaystyle m_a^2=\dfrac{c^2}{2}+\dfrac{b^2}{2}-\dfrac{a^2}{4}$

$\displaystyle m_b^2=\dfrac{c^2}{2}+\dfrac{a^2}{2}-\dfrac{b^2}{4}$

$\displaystyle m_c^2=\dfrac{a^2}{2}+\dfrac{b^2}{2}-\dfrac{c^2}{4}$

Where $a$, $b$, and $c$ are the sides opposite $A$, $B$, and $C$ respectively.

And $m_a$, $m_b$, and $m_c$ are the medians from $A$, $B$, and $C$ respectively.

Proof
We look at one of the medians, WLOG $m_c$:

We use Stewart's Theorem, noting that $\displaystyle AP=PB=\frac{c}{2}$ and $CP = m_c$

From Stewart's Theorem, this gives us

A similar argument can be used to show that the statement holds for the others medians.