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Math 2.0

Geometry Theorem

The medians meet at the centroid

Builds on Median of a triangle Centroid

Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$ , and let $G = \frac{1}{3}(A + B + C)$ be its centroid. Then $G$ lies on each of the three medians of $ABC$ . More precisely, let $M_A$ be the midpoint of $B$ and $C$ . Then $G = A + \frac{2}{3}(M_A - A)$ , and the same holds for the vertices $B$ and $C$ .

Remarks

The formula shows that $G$ divides each median in the ratio $2 : 1$ , counted from the vertex. The proof is one line: $A + \frac{2}{3}\left(\frac{B + C}{2} - A\right) = \frac{1}{3}(A + B + C)$ . The statement is true in each affine space over a field in which $2$ and $3$ are invertible. Mathlib states it for each simplex.

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Centroid concept
Median of a triangle concept
The medians meet at the centroid statement

Further reading

Web https://en.wikipedia.org/wiki/Centroid, Of a triangle
Web https://en.wikipedia.org/wiki/Median_(geometry), Relation to center of mass

About this page

Tag	`0GPQ67XV7N`
URI	`m2://0GPQ67XV7N`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04
Status	This copy of the site has no build data. Run `m2 status 0GPQ67XV7N`.
Uses	Median of a triangle, Centroid

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