Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

Equilateral triangle
TypeRegular polygon
Edges and vertices3
Schläfli symbol{3}
Coxeter–Dynkin diagrams
Symmetry groupD3
Area${\displaystyle {\tfrac {\sqrt {3}}{4}}a^{2}}$
Internal angle (degrees)60°

Principal properties

Denoting the common length of the sides of the equilateral triangle as ${\displaystyle a}$ , we can determine using the Pythagorean theorem that:

• The area is ${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2},}$
• The perimeter is ${\displaystyle p=3a\,\!}$
• The radius of the circumscribed circle is ${\displaystyle R={\frac {a}{\sqrt {3}}}}$
• The radius of the inscribed circle is ${\displaystyle r={\frac {\sqrt {3}}{6}}a}$  or ${\displaystyle r={\frac {R}{2}}}$
• The geometric center of the triangle is the center of the circumscribed and inscribed circles
• The altitude (height) from any side is ${\displaystyle h={\frac {\sqrt {3}}{2}}a}$

Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:

• The area of the triangle is ${\displaystyle A={\frac {3{\sqrt {3}}}{4}}R^{2}}$

Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:

• The area is ${\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}$
• The height of the center from each side, or apothem, is ${\displaystyle {\frac {h}{3}}}$
• The radius of the circle circumscribing the three vertices is ${\displaystyle R={\frac {2h}{3}}}$
• The radius of the inscribed circle is ${\displaystyle r={\frac {h}{3}}}$

In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.

Characterizations

A triangle ${\displaystyle ABC}$  that has the sides ${\displaystyle a}$ , ${\displaystyle b}$ , ${\displaystyle c}$ , semiperimeter ${\displaystyle s}$ , area ${\displaystyle T}$ , exradii ${\displaystyle r_{a}}$ , ${\displaystyle r_{b}}$ , ${\displaystyle r_{c}}$  (tangent to ${\displaystyle a}$ , ${\displaystyle b}$ , ${\displaystyle c}$  respectively), and where ${\displaystyle R}$  and ${\displaystyle r}$  are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.

Sides

• ${\displaystyle a=b=c}$
• ${\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}}$ [1]

Semiperimeter

• ${\displaystyle s=2R+\left(3{\sqrt {3}}-4\right)r}$ [2] (Blundon)
• ${\displaystyle s^{2}=3r^{2}+12Rr}$ [3]
• ${\displaystyle s^{2}=3{\sqrt {3}}T}$ [4]
• ${\displaystyle s=3{\sqrt {3}}r}$
• ${\displaystyle s={\frac {3{\sqrt {3}}}{2}}R}$

Angles

• ${\displaystyle A=B=C=60^{\circ }}$
• ${\displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}$
• ${\displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}}$ [5]

Area

• ${\displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad }$  (Weitzenböck)
• ${\displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}}$ [4]

• ${\displaystyle R=2r}$ [6] (Chapple-Euler)
• ${\displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}}$ [6]
• ${\displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}}$ [5]
• ${\displaystyle r_{a}=r_{b}=r_{c}}$

Equal cevians

Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:[7]

Coincident triangle centers

Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:

Six triangles formed by partitioning by the medians

For any triangle, the three medians partition the triangle into six smaller triangles.

• A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.[9]: Theorem 1
• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.[9]: Corollary 7

Points in the plane

• A triangle is equilateral if and only if, for every point ${\displaystyle P}$  in the plane, with distances ${\displaystyle p}$ , ${\displaystyle q}$ , and ${\displaystyle r}$  to the triangle's sides and distances ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle z}$  to its vertices,[10]: p.178, #235.4
${\displaystyle 4\left(p^{2}+q^{2}+r^{2}\right)\geq x^{2}+y^{2}+z^{2}.}$

Notable theorems

Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.

A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.[11]

Viviani's theorem states that, for any interior point ${\displaystyle P}$  in an equilateral triangle with distances ${\displaystyle d}$ , ${\displaystyle e}$ , and ${\displaystyle f}$  from the sides and altitude ${\displaystyle h}$ ,

${\displaystyle d+e+f=h,}$

independent of the location of ${\displaystyle P}$ .[12]

Pompeiu's theorem states that, if ${\displaystyle P}$  is an arbitrary point in the plane of an equilateral triangle ${\displaystyle ABC}$  but not on its circumcircle, then there exists a triangle with sides of lengths ${\displaystyle PA}$ , ${\displaystyle PB}$ , and ${\displaystyle PC}$ . That is, ${\displaystyle PA}$ , ${\displaystyle PB}$ , and ${\displaystyle PC}$  satisfy the triangle inequality that the sum of any two of them is greater than the third. If ${\displaystyle P}$  is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem.

Geometric construction

An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment.

An alternative method is to draw a circle with radius ${\displaystyle r}$ , place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.

In both methods a by-product is the formation of vesica piscis.

The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of Euclid's Elements.

Derivation of area formula

The area formula ${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}}$  in terms of side length ${\displaystyle a}$  can be derived directly using the Pythagorean theorem or using trigonometry.

Using the Pythagorean theorem

The area of a triangle is half of one side ${\displaystyle a}$  times the height ${\displaystyle h}$  from that side:

${\displaystyle A={\frac {1}{2}}ah.}$

The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base ${\displaystyle a}$ , and the hypotenuse is the side ${\displaystyle a}$  of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem

${\displaystyle \left({\frac {a}{2}}\right)^{2}+h^{2}=a^{2}}$

so that
${\displaystyle h={\frac {\sqrt {3}}{2}}a.}$

Substituting ${\displaystyle h}$  into the area formula ${\displaystyle {\frac {1}{2}}ah}$  gives the area formula for the equilateral triangle:

${\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}.}$

Using trigonometry

Using trigonometry, the area of a triangle with any two sides ${\displaystyle a}$  and ${\displaystyle b}$ , and an angle ${\displaystyle C}$  between them is

${\displaystyle A={\frac {1}{2}}ab\sin C.}$

Each angle of an equilateral triangle is 60°, so

${\displaystyle A={\frac {1}{2}}ab\sin 60^{\circ }.}$

The sine of 60° is ${\displaystyle {\tfrac {\sqrt {3}}{2}}}$ . Thus

${\displaystyle A={\frac {1}{2}}ab\times {\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{4}}ab={\frac {\sqrt {3}}{4}}a^{2}}$

since all sides of an equilateral triangle are equal.

Other properties

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

${\displaystyle {\frac {R}{r}}=2.}$

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]

${\displaystyle 3\left(p^{4}+q^{4}+t^{4}+a^{4}\right)=\left(p^{2}+q^{2}+t^{2}+a^{2}\right)^{2}.}$

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

${\displaystyle p^{2}+q^{2}+t^{2}=3\left(R^{2}+L^{2}\right),}$

${\displaystyle p^{4}+q^{4}+t^{4}=3\left[\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right],}$

where ${\displaystyle R}$  is the circumscribed radius and ${\displaystyle L}$  is the distance between point ${\displaystyle P}$  and the centroid of the equilateral triangle.

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]

${\displaystyle 4\left(p^{2}+q^{2}+t^{2}\right)=5a^{2},}$

${\displaystyle 16\left(p^{4}+q^{4}+t^{4}\right)=11a^{4}.}$

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]

${\displaystyle p=q+t,}$

${\displaystyle q^{2}+qt+t^{2}=a^{2}.}$

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

${\displaystyle z={\frac {t^{2}+tq+q^{2}}{t+q}},}$

which also equals ${\textstyle {\tfrac {t^{3}-q^{3}}{t^{2}-q^{2}}}}$  if ${\displaystyle t\neq q}$  and
${\displaystyle {\frac {1}{q}}+{\frac {1}{t}}={\frac {1}{y}},}$

which is the optic equation.

For an equilateral triangle:

• The ratio of its area to the area of the incircle, ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$ , is the largest of any triangle.[21]: Theorem 4.1
• The ratio of its area to the square of its perimeter, ${\displaystyle {\frac {1}{12{\sqrt {3}}}},}$  is larger than that of any non-equilateral triangle.[11]
• If a segment splits an equilateral triangle into two regions with equal perimeters and with areas ${\displaystyle A_{1}}$  and ${\displaystyle A_{2}}$ , then[10]: p.151, #J26

${\displaystyle {\frac {7}{9}}\leq {\frac {A_{1}}{A_{2}}}\leq {\frac {9}{7}}.}$

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

${\displaystyle z_{1}+\omega z_{2}+\omega ^{2}z_{3}=0.}$

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]

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

References

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