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. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
the altitude, a line segment from the apex perpendicular to the base,
the angle bisector from the apex to the base,
the median from the apex to the midpoint of the base,
the perpendicular bisector of the base within the triangle,
the segment within the triangle of the unique axis of symmetry of the triangle, and
the segment within the triangle of the Euler line of the triangle.
Video: Height of Isosceles Triangle
Their common length is the height of the triangle. If the triangle has equal sides of length and base of length , the general triangle formulas for the lengths of these segments all simplify to
This formula can also be derived from the Pythagorean theorem using the fact that .....
of the triangle. If the triangle has equal sides of length 
and base of length 
, the general triangle formulas for the lengths of these segments all simplify to
