Triangle

Definition

Let $K$ be a field and let $\mathcal{A}$ be an affine space over a $K$ -vector space. A triangle in $\mathcal{A}$ is an ordered triple $(A, B, C)$ of points of $\mathcal{A}$ that are not collinear. It is written $ABC$ or $\triangle ABC$ . The points $A$ , $B$ and $C$ are its vertices. The segments $BC$ , $CA$ and $AB$ are its sides.

Remarks

Three points are not collinear if and only if they are affinely independent: the vectors $B - A$ and $C - A$ are linearly independent. In a Euclidean space, the side lengths are often written $a = d(B, C)$ , $b = d(C, A)$ and $c = d(A, B)$ . The angles of the triangle are $\angle BAC$ , $\angle CBA$ and $\angle ACB$ . A triangle is isosceles if two sides have the same length, equilateral if all three do, and scalene if no two do. It is a right triangle if one angle is $\pi / 2$ . Mathlib's Affine.Triangle is an affinely independent family of three points, indexed by $\{0, 1, 2\}$ .

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Affine space concept
Collinear points concept
Triangle definition

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Orthocentric system

Tag	`0JDB8YJBSR`
URI	`m2://0JDB8YJBSR`
Kind	definition
Writing	W1 stated
MSC	51M04
Defines	Triangle
Uses	Collinear points, Affine space

Triangle

Definition

Remarks

Reading path

Where this leads

Further reading

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