Median Formula

Theorem
The length of a median is equal to

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

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

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

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 $$AP=PB=\frac{c}{2}$$ 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.