Equilateral triangle

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Equilateral triangle
Triangle.Equilateral.svg
Type Regular polygon
Edges and vertices 3
Schläfli symbol {3}
Coxeter–Dynkin diagrams 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 that has the sides , , , semiperimeter , area , exradii , , (tangent to , , respectively), and where and 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: [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.

Points in the plane

Notable theorems

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

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 in an equilateral triangle with distances , , and from the sides and altitude ,

independent of the location of . [12]

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

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 , 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 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 times the height 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 , and the hypotenuse is the side of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem

so that

Substituting into the area formula gives the area formula for the equilateral triangle:

Using trigonometry

Using trigonometry, the area of a triangle with any two sides and , and an angle 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.

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, . 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 to the inradius of any triangle, with [16] :p.198

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 is greater than or equal to 2, equality holding when 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 to the points where the angle bisectors of , , and cross the sides (, , and being the vertices). There are numerous other triangle inequalities that hold with equality if and only if the triangle is equilateral.

For any point in the plane, with distances , , and from the vertices , , and respectively, [18]

For any point in the plane, with distances , , and from the vertices, [19]

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

For any point on the inscribed circle of an equilateral triangle, with distances , , and from the vertices, [20]

For any point on the minor arc of the circumcircle, with distances , , and from , , and , respectively [12]

Moreover, if point on side divides into segments and with having length and having length , then [12] :172

which also equals if and

which is the optic equation.

For an equilateral triangle:

If a triangle is placed in the complex plane with complex vertices , , and , then for either non-real cube root of 1 the triangle is equilateral if and only if [22] :Lemma 2

The equilateral triangle tiling fills the plane. Tiling 3 simple.svg
The equilateral triangle tiling fills the plane.

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]

A regular tetrahedron is made of four equilateral triangles. Viervlak-frame.jpg
A regular tetrahedron is made of four equilateral triangles.

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 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 -simplexes, with . [28]

In culture and society

Equilateral triangles have frequently appeared in man made constructions:

See also

Related Research Articles

<span class="mw-page-title-main">Area</span> Size of a two-dimensional surface

Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary 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 analogue of the length of a curve or the volume of a solid . Two different regions may have the same area ; by synecdoche, "area" sometimes is used to refer to the region, as in a "polygonal area".

<span class="mw-page-title-main">Octahedron</span> Polyhedron with eight triangular faces

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

<span class="mw-page-title-main">Quadrilateral</span> Polygon with four sides and four corners

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

<span class="mw-page-title-main">Triangle</span> Shape with three sides

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

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

<span class="mw-page-title-main">Altitude (triangle)</span> Perpendicular line segment from a triangles side to opposite vertex

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the side opposite the vertex. 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.

<span class="mw-page-title-main">Incircle and excircles</span> Circles tangent to all three sides of a triangle

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

<span class="mw-page-title-main">Cyclic quadrilateral</span> 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.

<span class="mw-page-title-main">Isosceles triangle</span> Triangle with at least two sides congruent

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.

<span class="mw-page-title-main">Decagon</span> Shape with ten sides

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

<span class="mw-page-title-main">Gyroelongated square bipyramid</span> 17th Johnson solid

In geometry, the gyroelongated square bipyramid is a polyhedron with 16 triangular faces. it can be constructed from a square antiprism by attaching two equilateral square pyramids to each of its square faces. The same shape is also called hexakaidecadeltahedron, heccaidecadeltahedron, or tetrakis square antiprism; these last names mean a polyhedron with 16 triangular faces. It is an example of deltahedron, and of a Johnson solid.

<span class="mw-page-title-main">Snub disphenoid</span> Convex polyhedron with 12 triangular faces

In geometry, the snub disphenoid is a convex polyhedron with 12 equilateral triangles as its faces. It is an example of deltahedron and Johnson solid. It can be constructed in different approaches. This shape also has alternative names called Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron; these names mean the 12-sided polyhedron.

In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center.

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.

<span class="mw-page-title-main">Tangential quadrilateral</span> Polygon whose four sides all touch a circle

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. This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).

<span class="mw-page-title-main">Steiner inellipse</span> Unique ellipse tangent to all 3 midpoints of a given triangles sides

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.

<span class="mw-page-title-main">Pentagon</span> 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°.

<span class="mw-page-title-main">Bicentric quadrilateral</span> Convex, 4-sided shape with an incircle and a circumcircle

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.

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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 polychoron Pentachoron 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