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
\[\begin{aligned} m_a &= \frac{1}{2}\sqrt{2b^2+2c^2-a^2} \\ m_b &= \frac{1}{2}\sqrt{2a^2+2c^2-b^2} \\ m_c &= \frac{1}{2}\sqrt{2a^2+2b^2-c^2}\end{aligned}\]
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
\[\begin{aligned} a &= \frac{2}{3}\sqrt{2m_b^2 + 2m_c^2 -m_a^2} \\ b &= \frac{2}{3}\sqrt{2m_a^2 + 2m_c^2 -m_b^2} \\ c &= \frac{2}{3}\sqrt{2m_a^2 + 2m_b^2 - m_c^2} \end{aligned}\]
Proof
TODO
Note
This means that a triangle is completely determined by its medians.