Equilateral triangle

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 the familiar 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

An equilateral triangle. It has equal sides (a=b=c), equal angles ($\alpha = \beta =\gamma$), and equal altitudes (h_a=h_b=h_c).

Denoting the common length of the sides of the equilateral triangle as 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 $R = \frac{a}{\sqrt{3}}$
• The radius of the inscribed circle is $r=\frac{\sqrt{3}}{6} a$ or $r=\frac{R}{2}$
• The geometric center of the triangle is the center of the circumscribed and inscribed circles
• And the altitude (height) from any side is $h=\frac{\sqrt{3}}{2} a$.

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

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

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

• The area is $A=\frac{h^2}{\sqrt{3}}$
• The height of the center from each side is $\frac{h}{3}$
• 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 r_a, r_b, r_c (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 nine categories is true. Thus these are properties that are unique to equilateral triangles.

Sides

• $\displaystyle a=b=c$
• $\displaystyle a^2+b^2+c^2=ab+bc+ca$ ^[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

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

• The three altitudes have equal lengths.
• The three medians have equal lengths.
• The three angle bisectors have equal lengths.

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]:p.37
• 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}

Points in the plane

• A triangle is equilateral if and only if, for every point P in the plane, with distances p, q, and r to the triangle's sides and distances x, y, and z to its vertices,^{[11]:p.178,#235.4}

$4(p^2+q^2+r^2) \geq x^2+y^2+z^2.$

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.^[12]

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.^[13]

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.^[14]:p.198

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.^[15]

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.^{[16]:Theorem 4.1}

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.^[12]

If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A₁ and A₂ , then^{[11]:p.151,#J26}

$\frac{7}{9} \leq \frac{A_1}{A_2} \leq \frac{9}{7}.$

If a triangle is placed in the complex plane with complex vertices z₁, z₂, and z₃, then for either non-real cube root $\omega$ of 1 the triangle is equilateral if and only if^{[17]:Lemma 2}

$z_1+\omega z_2+\omega^2 z_3 =0.$

Given a point 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 or equals 2, equality hold when P is the centroid and is less than that of any other triangle.^[18] 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 P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides (A, B, and C being the vertices).

For any point P in the plane, with distances p, q, and t from the vertices A, B, and C respectively,^[19]

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

For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices,^[19]

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

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,^[13]:170[19]

$\displaystyle p=q+t$

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, then^[13]:172

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

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

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

which is the optic equation.

There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral.

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, as 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 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 $A = \frac{\sqrt{3}}{4}a^2$ in terms of side length 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 a times the height h from that side:

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

An equilateral triangle with a side of 2 has a height of √3 as the sine of 60° is √3/2.

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

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

so that

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

Substituting h into the area formula (1/2)ah gives the area formula for the equilateral triangle:

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

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.

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

• Some archaeological sites have equilateral triangles as part of their construction, for example Lepenski Vir in Serbia.
• The shape also occurs in modern architecture such as Randhurst Mall and the Jefferson National Expansion Memorial.
• The Flag of the Philippines, the Seal of the President of the Philippines and the Flag of Junqueirópolis contain equilateral triangles.
• It is a shape of a variety of road signs, including the Yield sign.
• The fraternity Tau Kappa Epsilon uses the equilateral triangle as its primary symbol.

See also

• Almost-equilateral Heronian triangle
• Isosceles triangle
• Right triangle
• Trigonometry

References

1. Andreescu, Titu; Andrica, Dorian (2006). Complex Numbers from A to...Z. Birkhäuser. pp. 70, 113–115.
2. Pohoata, Cosmin (2010). "A new proof of Euler's inradius - circumradius inequality" (PDF). Gazeta Matematica Seria B (3): 121–123.
3. Bencze, Mihály; Wu, Hui-Hua; Wu, Shan-He (2008). "An equivalent form of fundamental triangle inequality and its applications" (PDF). Research Group in Mathematical Inequalities and Applications 11 (1).
4. Dospinescu, G.; Lascu, M.; Pohoata, C.; Letiva, M. (2008). "An elementary proof of Blundon's inequality" (PDF). Journal of inequalities in pure and applied mathematics 9 (4).
5. Blundon, W. J. (1963). "On Certain Polynomials Associated with the Triangle". Mathematics Magazine 36 (4): 247–248. doi:10.2307/2687913.
6. Alsina, Claudi; Nelsen, Roger B. (2009). When less is more. Visualizing basic inequalities. Mathematical Association of America. pp. 71, 155.
7. McLeman, Cam; Ismail, Andrei. "Weizenbock's inequality". PlanetMath.
8. Owen, Byer; Felix, Lazebnik; Deirdre, Smeltzer (2010). Methods for Euclidean Geometry. Mathematical Association of America. pp. 36, 39.
9. Yiu, Paul (1998). "Notes on Euclidean Geometry" (PDF).
10. Cerin, Zvonko (2004). "The vertex-midpoint-centroid triangles" (PDF). Forum Geometricorum 4: 97–109.
11. "Inequalities proposed in "Crux Mathematicorum"" (PDF).
12. Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
13. Posamentier, Alfred S.; Salkind, Charles T. (1996). Challenging Problems in Geometry. Dover Publ.
14. Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities" (PDF). Forum Geometricorum 12: 197–209.
15. Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover Publ. pp. 379–380.
16. Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials". American Mathematical Monthly 115 (October): 679–689.
17. Dao, Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers" (PDF). Forum Geometricorum 15: 105–114.
18. Lee, Hojoo (2001). "Another proof of the Erdős–Mordell Theorem" (PDF). Forum Geometricorum 1: 7–8.
19. De, Prithwijit (2008). "Curious properties of the circumcircle and incircle of an equilateral triangle". Mathematical Spectrum 41 (1): 32–35.

External links

• Weisstein, Eric W., "Equilateral Triangle", MathWorld.