Equilateral Triangle - Other Properties

Other Properties

By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2.

The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.

The ratio of the area of the incircle to the area of an equilateral triangle, is larger than that of any non-equilateral triangle.

The ratio of the area to the square of the perimeter of an equilateral triangle, is larger than that for any other triangle.

Given a point in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides equals 2 and is less than that of any other triangle. This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances to the points where the angle bisectors cross the sides.

For any point P in the plane, with distances p, q, and t from the vertices A, B, and C respectively,

For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices,

and

For any point P on the minor arc BC of the circumcircle, with distances p, q, and t from A, B, and C respectively,

and

moreover, if point D on side BC divides PA into segments PD and DA with DA having length z and PD having length y, then

which also equals if t ≠ q; and

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the dihedral group of order 6 D₃.

Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle).

Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.