Is an equilateral triangle always isosceles?

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length

and the remaining side has length

. This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos (leg).

A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal. Another special case of an isosceles triangle is the isosceles right triangle.

The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as

The area is therefore given by

The inradius of an isosceles triangle is given by

The mean of

is given by

so the geometric centroid is

or 2/3 the way from its vertex (Gearhart and Schulz 1990).

Considering the angle at the apex of the triangle and writing

instead of

, there is a surprisingly simple relationship between the area and vertex angle . As shown in the above diagram, simple trigonometry gives

so the area is

Erecting similar isosceles triangles on the edges of an initial triangle

gives another triangle

such that

, and

concur. The triangles are therefore perspective triangles.

No set of

points in the plane can determine only isosceles triangles.

30-60-90 Triangle, Acute Triangle, Equilateral Triangle, Golden Gnomon, Golden Triangle, Isosceles Right Triangle, Isosceles Tetrahedron, Isoscelizer, Kiepert Parabola, Obtuse Triangle, Petr-Neumann-Douglas Theorem, Point Picking, Pons Asinorum, Right Triangle, Scalene Triangle, Steiner-Lehmus Theorem Explore this topic in the MathWorld classroom Gearhart, W. B. and Schulz, H. S. "The Function

." College Math. J. 21, 90-99, 1990.

Weisstein, Eric W. "Isosceles Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsoscelesTriangle.html