Triangles
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
A
,
{\displaystyle A,}
B
,
{\displaystyle B,}
and
C
{\displaystyle C}
is denoted
△
A
B
C
.
{\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
a
{\displaystyle a}
be the length of the edge
B
C
,
{\displaystyle BC,}
b
{\displaystyle b}
the length of
C
A
,
{\displaystyle CA,}
and
c
{\displaystyle c}
the length of
A
B
{\displaystyle AB}
; and to represent the angle measure at each corner using a Greek letter, letting
α
{\displaystyle \alpha }
be the measure of angle
∠
C
A
B
,
{\displaystyle \angle CAB,}
β
{\displaystyle \beta }
the measure of
∠
A
B
C
,
{\displaystyle \angle ABC,}
and
γ
{\displaystyle \gamma }
the measure of
∠
B
C
A
.
{\displaystyle \angle BCA.}
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