Isosceles triangle

A triangle with two equal sides; the equal sides are the legs and the third is the base, the angle between the legs is the apex angle and the two at the base are the base angles, which are necessarily equal.

Dependencies: Triangle, Distance. A derived definition: add to a triangle the condition that two sides have equal length.

Definition

Let $\triangle ABC$ be a triangle.

An isosceles triangle with vertex angle at A and equal-side marks on AB and AC.

If two sides have equal length, for example $|AB|=|AC|$ , the triangle is called isosceles. The equal sides $AB$ and $AC$ are its legs, the third side $BC$ is its base, $\angle A$ is the vertex angle, and $\angle B,\angle C$ are the base angles.

One equality of side lengths produces a whole family of symmetries: equal base angles, and a vertex-angle bisector that is also a median and an altitude.

Remarks

Equal sides and equal angles. Equal legs imply equal base angles (Base angles of an isoceles triangle are equal), and equal base angles imply equal opposite sides (Isoceles converse: equal base angles ⇒ equal sides).
Three lines coincide. The vertex angle bisector, the median to the base, and the altitude to the base are the same segment (Isoceles: median, altitude, bisector coincide).
An equilateral triangle is a special case. With all three sides equal, the triangle is isosceles relative to any choice of base, and all three angles are $60^\circ$ .

Definition

Remarks