Isosceles

Isosceles triangle
Isosceles triangle with vertical axis of symmetry
Type	triangle
Edges and vertices	3
Schläfli symbol	( ) ∨ { }
Symmetry group	Dih2, [ ], (*), order 2
Dual polygon	Self-dual
Properties	convex, cyclic

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.

By the isosceles triangle theorem, the two angles opposite the equal sides are themselves equal, while if the third side is different then the third angle is different.

By the Steiner–Lehmus theorem, every triangle with two angle bisectors of equal length is isosceles.

In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles. The vertex opposite the base is called the apex.

Euclid defined an isosceles triangle as one having exactly two equal sides, but modern treatments prefer to define them as having at least two equal sides, making equilateral triangles (with three equal sides) a special case of isosceles triangles. In the equilateral triangle case, since all sides are equal, any side can be called the base, if needed, and the term leg is not generally used.

A triangle with exactly two equal sides has exactly one axis of symmetry, which goes through the vertex angle and also goes through the midpoint of the base. Thus the axis of symmetry coincides with (1) the angle bisector of the vertex angle, (2) the median drawn to the base, (3) the altitude drawn from the vertex angle, and (4) the perpendicular bisector of the base.