Apollonius's theorem

Statement

Let $A$, $B$ and $C$ be points of the Euclidean space $\mathbb{R}^n$, and let $D$ be the midpoint of $B$ and $C$. Let $|XY|$ denote the distance between the points $X$ and $Y$. Then $$|AB|^2 + |AC|^2 = 2 \left( |AD|^2 + |BD|^2 \right).$$

Remarks

For a triangle $ABC$ with the side lengths $a = |BC|$, $b = |CA|$, $c = |AB|$, the median from $A$ thus has the length $m_a = \frac{1}{2} \sqrt{2 b^2 + 2 c^2 - a^2}$. In vector form, with $u = B - A$ and $v = C - A$, the statement is the parallelogram law $\|u + v\|^2 + \|u - v\|^2 = 2\|u\|^2 + 2\|v\|^2$. For an isosceles triangle with $|AB| = |AC|$, the median $AD$ is perpendicular to $BC$, and the theorem reduces to the Pythagorean theorem.

History

The theorem is Proposition 122 of Book VII of the Collection of Pappus of Alexandria. It was possibly in the lost work Plane Loci of Apollonius of Perga.

Explanations

L1 human

The theorem relates the two sides from $A$ to the median from $A$. If you know the three sides of a triangle, it gives the length of each median. For the right triangle with legs $3$, $4$ and hypotenuse $5$, the median to the hypotenuse has length $\frac{1}{2}\sqrt{18 + 32 - 25} = \frac{5}{2}$.