Medians

Definition: Median

A median in a triangle is a cevian which connects a vertex of the triangle to the midpoint of the opposite side.

NOTATION

The median towards the side $s$ is usually denoted as $m_s$.

Properties

Theorem: Concurrency of a Triangle's Medians

The medians of a triangle are all concurrent and intersect at its centroid.

PROOF

TODO

Theorem: Median Lengths from Side Lengths (Apollonius's Theorem)

If a triangle has sides $a,b,c$, then their respective medians are given by

\begin{align*} m_a &= \frac{1}{2}\sqrt{2b^2+2c^2-a^2} \\ m_b &= \frac{1}{2}\sqrt{2a^2+2c^2-b^2} \\ m_c &= \frac{1}{2}\sqrt{2a^2+2b^2-c^2}\end{align*}

PROOF

TODO

Theorem: Side Lengths from Median Lengths

If a triangle has medians $m_a,m_b,m_c$, then their respective sides are given by

\begin{align*} a &= \frac{2}{3}\sqrt{2m_b^2 + 2m_c^2 -m_a^2} \\ b &= \frac{2}{3}\sqrt{2m_a^2 + 2m_c^2 -m_b^2} \\ c &= \frac{2}{3}\sqrt{2m_a^2 + 2m_b^2 - m_c^2} \end{align*}

PROOF

TODO

NOTE

This means that a triangle is completely determined by its medians.