triangle buildings

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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