# Equilateral triangle

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Equilateral triangle
Type Regular polygon
Edges and vertices 3
Schläfli symbol {3}
Coxeter diagram
Symmetry group D3
Area $\tfrac{\sqrt{3}}{4} a^2$
Internal angle (degrees) 60°

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.

## Principal properties

Equilateral triangle

Assuming the lengths of the sides of the equilateral triangle are a, we can determine using the Pythagorean theorem that:

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

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

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

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

## Characterizations

A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following eight categories is true. These are also properties of an equilateral triangle.

### Sides

• $\displaystyle a^2+b^2+c^2=ab+bc+ca.$1
• $\displaystyle abc=(a+b-c)(a-b+c)(-a+b+c)\quad\text{(Lehmus)}$2
• $\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{\sqrt{25Rr-2r^2}}{4Rr}.$3

### Semiperimeter

• $\displaystyle s=2R+(3\sqrt{3}-4)r\quad\text{(Blundon)}$4
• $\displaystyle s^2=3r^2+12Rr.$5
• $\displaystyle s^2=3\sqrt{3}T.$6
• $\displaystyle s=3\sqrt{3}r$
• $\displaystyle s=\frac{3\sqrt{3}}{2}R$

### 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}.$2

### Area

• $\displaystyle T=\frac{a^2+b^2+c^2}{4\sqrt{3}}\quad\text{(Weizenbock)}$7
• $\displaystyle T=\frac{\sqrt{3}}{4}(abc)^{^{\frac{2}{3}}}.$6

### Circumradius, inradius and exradii

• $\displaystyle R=2r\quad\text{(Chapple-Euler)}$1
• $\displaystyle 9R^2=a^2+b^2+c^2.$1
• $\displaystyle r=\frac{r_a+r_b+r_c}{9}.$2
• $\displaystyle r_a=r_b=r_c.$

### Equal cevians

Three kinds of cevians are equal for (and only for) equilateral triangles:8

### Coincident triangle centers

Every triangle center of an equilateral triangle coincides with its centroid, and for some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular, a triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide.9 It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides with its nine-point center.1

### 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.10: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.10:Corollary 7

## Famous 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 P in an equilateral triangle, with distances d, e, and f from the sides, d + e + f = the altitude of the triangle, independent of the location of P.12

Pompeiu's theorem states that, if P is an arbitrary point in an equilateral triangle ABC, then there exists a triangle with sides of length PA, PB, and PC.

## 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.13

The ratio of the area of the incircle to the area of an equilateral triangle, $\frac{\pi}{3\sqrt{3}}$, is larger than that of any non-equilateral triangle.14

The ratio of the area to the square of the perimeter of an equilateral triangle, $\frac{1}{12\sqrt{3}},$ is larger than that for any other triangle.11

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.15 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,16

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

For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices,16

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

and

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

For any point P on the minor arc BC of the circumcircle, with distances p, q, and t from A, B, and C respectively,16

$\displaystyle p=q+t$

and

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

moreover, if point D on side BC divides PA into segments PD and DA with DA having length z and PD having length y, then12

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

which also equals $\tfrac{t^{3}-q^{3}}{t^{2}-q^{2}}$ if tq; and

$\frac{1}{q}+\frac{1}{t}=\frac{1}{y}.$

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).

A regular tetrahedron is made of four equilateral triangles.

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.

## Geometric construction

Construction of equilateral triangle with compass and straightedge

An equilateral triangle is easily constructed using a compass and straightedge. 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

Alternate method:

Draw a circle with radius 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.

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

## In culture and society

Equilateral triangles have frequently appeared in man made constructions:

## Derivation of area formula

### Using the Pythagorean theorem

The area of a triangle is half the base a times the height h,

$A = \frac{1}{2} ah.$
An equilateral triangle with a side of 2 has a height of 3 and the sine of 60° is 3/2.

The legs of the right triangle are half of the base and the height, while the hypotenuse is the side or base of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem

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

The equation can be rearranged to find the square of the height

$h^2 = a^2 - \left(\frac{a}{2}\right)^2.$

The height is the square root of the equation. Thus

$h = \sqrt{a^2 - \left(\frac{a}{2}\right)^2} = \sqrt{a^2 - \frac{a^2}{4}} = \sqrt{\frac{3}{4}a^2} = \frac{\sqrt{3}}{2}a.$

By substituting $h = {\tfrac{\sqrt{3}}{2}a}$, the formula for the area of an equilateral triangle can be derived according to

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

### Using trigonometry

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

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

Each angle of an equilateral triangle is 60°, so

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

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

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

since all sides of an equilateral triangle are equal.

## References

1. ^ a b c d Andreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 70, 113-115.
2. ^ a b c Pohoata, Cosmin, "A new proof of Euler's inradius - circumrdius inequality", Gazeta Matematica Seria B, no. 3, 2010, pp. 121-123, [1].
3. ^ M. Bencze, Hui-Hua Wu and Shan-He Wu, "An equivalent form of fundamental triangle inequality and its applications", Research Group in Mathematical Inequalities and Applications, Volume 11, Issue 1, 2008, [2]
4. ^ G. Dospinescu, M. Lascu, C. Pohoata & M. Letiva, "An elementary proof of Blundon's inequality", Journal of inequalities in pure and applied mathematics, vol. 9, iss. 4, 2008, [3]
5. ^ Blundon, W. J., "On Certain Polynomials Associated with the Triangle", Mathematics Magazine, Vol. 36, No. 4 (Sep., 1963), pp. 247-248.
6. ^ a b Alsina, Claudi & Nelsen, Roger B., When less is more. Visualizing basic inequalities, Mathematical Association of America, 2009, pp. 71, 155.
7. ^ Cam McLeman & Andrei Ismail, "Weizenbock's inequality", PlanetMath, [4].
8. ^ Byer, Owen; Lazebnik, Felix and Smeltzer, Deirdre, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 36, 39.
9. ^ Yiu, Paul, Notes on Euclidean Geometry, 1998, p. 37, [5]
10. ^ a b Cˇerin, Zvonko, "The vertex-midpoint-centroid triangles", Forum Geometricorum 4, 2004: pp. 97–109.
11. ^ a b Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
12. ^ a b Posamentier, Alfred S., and Salkind, Charles T., Challenging Problems in Geometry, Dover Publ., 1996.
13. ^ Dorrie, Heinrich, 100 Great Problems of Elementary Mathematics, Dover Publ., 1965: 379-380.
14. ^ Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", American Mathematical Monthly 115, October 2008, 679-689: Theroem 4.1.
15. ^ Lee, Hojoo, "Another proof of the Erdős–Mordell Theorem", Forum Geometricorum 1, 2001: 7-8.
16. ^ a b c De, Prithwijit, "Curious properties of the circumcircle and incircle of an equilateral triangle," Mathematical Spectrum 41(1), 2008-2009, 32-35.

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