Equilateral triangle

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Equilateral triangle
Triangle.Equilateral.svg
Type Regular polygon
Edges and vertices 3
Schläfli symbol {3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.png
Symmetry group D3
Area
Internal angle (degrees)60°

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.

Contents

Principal properties

An equilateral triangle. It has equal sides (
a
=
b
=
c
{\displaystyle a=b=c}
), equal angles (
a
=
b
=
g
{\displaystyle \alpha =\beta =\gamma }
), and equal altitudes (
h
a
=
h
b
=
h
c
{\displaystyle h_{a}=h_{b}=h_{c}}
). Equilateral-triangle-heights.svg
An equilateral triangle. It has equal sides (), equal angles (), and equal altitudes ().

Denoting the common length of the sides of the equilateral triangle as , we can determine using the Pythagorean theorem that:

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

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

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 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

Semiperimeter

Angles

Area

Circumradius, inradius, and exradii

Equal cevians

Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles: [8]

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.

Points in the plane

Notable theorems

Visual proof of Viviani's theorem
1.
Nearest distances from point P to sides of equilateral triangle ABC are shown.
2.
Lines DE, FG, and HI parallel to AB, BC and CA, respectively, define smaller triangles PHE, PFI and PDG.
3.
As these triangles are equilateral, their altitudes can be rotated to be vertical.
4.
As PGCH is a parallelogram, triangle PHE can be slid up to show that the altitudes sum to that of triangle ABC. Viviani theorem visual proof.svg
Visual proof of Viviani's theorem
1.Nearest distances from point P to sides of equilateral triangle ABC are shown.
2.Lines DE, FG, and HI parallel to AB, BC and CA, respectively, define smaller triangles PHE, PFI and PDG.
3.As these triangles are equilateral, their altitudes can be rotated to be vertical.
4.As PGCH is a parallelogram, triangle PHE can be slid up to show that the altitudes sum to that of triangle ABC.

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 and altitude h,

independent of the location of P. [13]

Pompeiu's theorem states that, if P is an arbitrary point in the plane of an equilateral triangle ABC but not on its circumcircle, then there exists a triangle with sides of lengths PA, PB, and PC. That is, PA, PB, and PC satisfy the triangle inequality that the sum of any two of them is greater than the third. If 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.

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, , 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, 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

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

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 than or equal to 2, equality holding when P is the centroid. In no other triangle is there a point for which this ratio is as small as 2. [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]

For any point P in the plane, with distances p, q, and t from the vertices, [20]

and

where R is the circumscribed radius and L is the distance between point P and the centroid of the equilateral triangle.

For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices, [21]

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, [13]

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 [13] :172

which also equals if tq; and

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).

The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees. [22]

The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one). [23] :p. 19

A regular tetrahedron is made of four equilateral triangles. Viervlak-frame.jpg
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 Equilateral triangle construction.svg
Construction of equilateral triangle with compass and straightedge

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 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.

Equilateral Triangle Inscribed in a Circle.gif

Derivation of area formula

The area formula 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:

An equilateral triangle with a side of 2 has a height of [?]3, as the sine of 60deg is [?]3/2. Equilateral triangle with height square root of 3.svg
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

so that

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

Using trigonometry

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

Each angle of an equilateral triangle is 60°, so

The sine of 60° is . Thus

since all sides of an equilateral triangle are equal.

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

See also

Related Research Articles

Area Extent of a two-dimensional surface

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve or the volume of a solid.

Quadrilateral polygon with four sides and four corners

In Euclidean plane geometry, a quadrilateral is a polygon with four edges (sides) and four vertices (corners). Other names for quadrilateral include quadrangle, tetragon, and 4-gon. A quadrilateral with vertices , , and is sometimes denoted as .

Triangle Shape with three sides

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .

Right triangle

A right triangle or right-angled triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

Hexagon Shape with a few sides

In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.

Incircle and excircles of a triangle Circles tangent to all three sides of a triangle

In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

Cyclic quadrilateral Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In geometry, three or more than three straight lines make a polygon and an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, it doesn’t need to be equiangular, but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon doesn’t need to be a convex polygon: it could be concave or even self-intersecting.

Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

Ptolemys theorem Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

Circumscribed circle

In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.

In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly.

Tangential quadrilateral

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős (1935) posed the problem of proving the inequality; a proof was provided two years later by Mordell and D. F. Barrow (1937). This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).

Steiner inellipse

In geometry, the Steiner inellipse, midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inellipse. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.

Pentagon shape with five sides

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

Orthodiagonal quadrilateral

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

Acute and obtuse triangles

An acute triangle is a triangle with three acute angles. An obtuse triangle is a triangle with one obtuse angle and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.

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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds