Equilateral triangle

An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center, whose symmetry group is the dihedral group of order 6, ${\displaystyle \mathrm {D} _{3}}$. The integer-sided equilateral triangle is the only triangle with integer sides, and three rational angles as measured in degrees.^[13] It is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes),^{[14]: p. 19} and the only triangle whose Steiner inellipse is a circle (specifically, the incircle). 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 also equilateral.^[15] It is the only regular polygon aside from the square that can be inscribed inside any other regular polygon.


By Euler's inequality, the equilateral triangle has the smallest ratio of the circumradius ${\displaystyle R}$ to the inradius ${\displaystyle r}$ of any triangle, with^{[16]: p.198}


Given a point ${\displaystyle P}$ 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 is greater than or equal to 2, equality holding when ${\displaystyle P}$ is the centroid. In no other triangle is there a point for which this ratio is as small as 2.^[17] 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 from ${\displaystyle P}$ to the points where the angle bisectors of ${\displaystyle \angle APB}$, ${\displaystyle \angle BPC}$, and ${\displaystyle \angle CPA}$ cross the sides (${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ being the vertices). There are numerous other triangle inequalities that hold with equality if and only if the triangle is equilateral.


For any point ${\displaystyle P}$ in the plane, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ respectively,^[18]


For any point ${\displaystyle P}$ in the plane, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices,^[19]


For any point ${\displaystyle P}$ on the inscribed circle of an equilateral triangle, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from the vertices,^[20]


For any point ${\displaystyle P}$ on the minor arc ${\displaystyle BC}$ of the circumcircle, with distances ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle t}$ from ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, respectively^[12]


Moreover, if point ${\displaystyle D}$ on side ${\displaystyle BC}$ divides ${\displaystyle PA}$ into segments ${\displaystyle PD}$ and ${\displaystyle DA}$ with ${\displaystyle DA}$ having length ${\displaystyle z}$ and ${\displaystyle PD}$ having length ${\displaystyle y}$, then^[12]: 172


For an equilateral triangle:


If a triangle is placed in the complex plane with complex vertices ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, and ${\displaystyle z_{3}}$, then for either non-real cube root ${\displaystyle \omega }$ of 1 the triangle is equilateral if and only if^{[22]: Lemma 2}


Notably, the equilateral triangle tiles two dimensional space with six triangles meeting at a vertex, whose dual tessellation is the hexagonal tiling. 3.12², 3.4.6.4, (3.6)², 3².4.3.4, and 3⁴.6 are all semi-regular tessellations constructed with equilateral triangles.^[23]


In three dimensions, equilateral triangles form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles: the tetrahedron, octahedron and icosahedron.^{[24]: p.238} In particular, the tetrahedron, which has four equilateral triangles for faces, can be considered the three-dimensional analogue of the triangle. All Platonic solids can inscribe tetrahedra, as well as be inscribed inside tetrahedra. Equilateral triangles also form uniform antiprisms as well as uniform star antiprisms in three-dimensional space. For antiprisms, two (non-mirrored) parallel copies of regular polygons are connected by alternating bands of ${\displaystyle 2n}$ equilateral triangles.^[25] Specifically for star antiprisms, there are prograde and retrograde (crossed) solutions that join mirrored and non-mirrored parallel star polygons.^[26][27] The Platonic octahedron is also a triangular antiprism, which is the first true member of the infinite family of antiprisms (the tetrahedron, as a digonal antiprism, is sometimes considered the first).^{[24]: p.240}


As a generalization, the equilateral triangle belongs to the infinite family of ${\displaystyle n}$-simplexes, with ${\displaystyle n=2}$.^[28]

Equilateral triangles have frequently appeared in man made constructions: