Median length formula (Apollonius)

Dependencies: theorems Pythagorean theorem, Parallelogram properties (opposite sides equal, diagonals bisect each other), Parallelogram tests.

Statement

Let $\triangle ABC$ have side lengths $a=|BC|$, $b=|CA|$, $c=|AB|$. Let $M$ be the midpoint of $BC$, and denote by $m_a=|AM|$ the length of the median from $A$. Then

By symmetry, the median $m_b$ from $B$ to the midpoint of $AC$ and the median $m_c$ from $C$ to the midpoint of $AB$ each satisfy a formula of the same shape (just relabel).