Apollonius's theorem

Builds on Median of a triangle Midpoint Euclidean distance

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.

Reading path

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Tag	`0NAKHQ9XYY`
URI	`m2://0NAKHQ9XYY`
Kind	statement, theorem
Writing	W1 stated
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0NAKHQ9XYY`.
Uses	Median of a triangle, Midpoint, Euclidean distance
Same as	Wikidata Q877489 Wikipedia Apollonius's theorem

Apollonius's theorem

Statement

Remarks

History

Reading path

Further reading

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