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 .....
In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid.
Video: What is an Isosceles Trapezoid?
Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).
Special cases
Special cases of isosceles trapezoids
Rectangles and squares are usually considered to be special cases of isosceles trapezoids though some sources would exclude them.
Another special case is a 3-equal side trapezoid, sometimes known as a trilateral trapezoid or a trisosceles trapezoid. They can also be seen dissected from regular polygons of 5 sides or more as a truncation of 4 sequential vertices.
Video: Quadrilaterals - Trapezoids, Parallelograms, Rectangles, Squares, and Rhombuses
Self-intersections
Any non-self-crossing quadrilateral with exactly one axis of symmetry must be either an isosceles trapezoid or a kite. However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the antiparallelograms, crossed quadrilaterals in which opposite sides have equal length.
Every antiparallelogram has an isosceles trapezoid as its convex hull, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.
Convex isosceles trapezoid
Crossed isosceles trapezoid
antiparallelogram
Characterizations
If a quadrilateral is known to be a trapezoid, it is not necessary to check that the legs have the same length in order to know that it is an isosceles trapezoid (nor, under the definitions given in Wikipedia, is it sufficient, since a rhombus is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides); any of the following properties also distinguishes an isosceles trapezoid from other trapezoids:
The diagonals have the same length.
The base angles have the same measure.
The segment that joins the midpoints of the parallel sides is ....
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
