Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points, when non-collinear, determine a unique triangle situated within a unique flat plane. More generally, several points in Euclidean space of arbitrary dimension determine a simplex.

In non-Euclidean geometries three straight segments also determine a triangle, for instance a spherical triangle or hyperbolic triangle. A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides which are straight relative to the surface. A curvilinear triangle is a shape with three curved sides, for instance a circular triangle with circular-arc sides. This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.

A triangle with vertices ${\displaystyle A,}$ ${\displaystyle B,}$ and ${\displaystyle C}$ is denoted ${\displaystyle \triangle ABC.}$ In describing metrical relations within a triangle, it is common to represent the length of the edge opposite each vertex using a lower-case letter, letting ${\displaystyle a}$ be the length of the edge ${\displaystyle BC,}$ ${\displaystyle b}$ the length of ${\displaystyle CA,}$ and ${\displaystyle c}$ the length of ${\displaystyle AB}$; and to represent the angle measure at each corner using a Greek letter, letting ${\displaystyle \alpha }$ be the measure of angle ${\displaystyle \angle CAB,}$ ${\displaystyle \beta }$ the measure of ${\displaystyle \angle ABC,}$ and ${\displaystyle \gamma }$ the measure of ${\displaystyle \angle BCA.}$