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 $\{0, 1, 2\}$.

Explanations

L1 human

Pick three points that are not on one straight line and join each pair. The result is a triangle. If the three points were on one line, the "triangle" would be flat, and it would have no area.