Definition:Triangle (Geometry)/Isosceles

Definition

An isosceles triangle is a triangle in which two sides are the same length.

Base

The base of an isosceles triangle is specifically defined to be the side which is a different length from the other two.

In the above diagram, $BC$ is the base.

Base Angles

The two (equal) vertices adjacent to the base of an isosceles triangle are called the base angles.

In the above diagram, $\angle ABC$ and $\angle ACB$ are the base angles.

Apex

The vertex opposite the base of an isosceles triangle is called the apex of the triangle.

In the above diagram, $A$ is the apex.

Legs

The sides of an isosceles triangle which are adjacent to the apex are called the legs of the triangle.

In the above diagram, $AB$ and $AC$ are the legs.

Also defined as

Some sources require that in an isosceles triangle, the third side has to be specifically of a different length than the two defined as being equal.

Hence, under such a definition, an equilateral triangle would not be classified as isosceles.

Euclid's Definition

In the words of Euclid:

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

(The Elements: Book $\text{I}$: Definition $20$)

Also see

• Results about isosceles triangles can be found here.

Linguistic Note

The word isosceles comes from the Greek: $\iota \sigma \omicron \sigma \kappa \epsilon \lambda \epsilon \varsigma$, that is: from iso meaning equal, and skelos meaning leg.

Thus an isosceles triangle is literally an equal-leg triangle.

It is pronounced eye-sos-ell-eez, that is, with the emphasis on the second syllable. Note that the c is silent.

The word skeleton comes from the same linguistic root.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): triangle (Euclidean geometry)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): isosceles triangle
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): triangle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): isosceles triangle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): triangle
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): isosceles triangle