Equilateral triangle

"Equilateral" redirects here. For other uses, see Equilateral (disambiguation).

Equilateral triangle
Type	Regular polygon
Edges and vertices	3
Schläfli symbol	{3}
Symmetry group	$\mathrm {D} _{3}$
Area	${\textstyle {\frac {\sqrt {3}}{4}}a^{2}}$
Internal angle (degrees)	60°

An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.

The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry.

Definition and characterizations

A triangle is said to be equilateral if and only if it has three equal sides and three equal internal angles; those angles are 60°. An equilateral triangle is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. Consequently, since the perimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side $p=3\ell$. Because of these characterizations, the equilateral triangles are regular polygons. The cevians of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on the base's choice. The equilateral triangle is symbolically denoted as $\{3\}$ by Schläfli notation.

Properties

Area

The right triangle with a hypotenuse of $1$ has a height of ${\sqrt {3}}/2$, the sine of 60°.

The area of an equilateral triangle with edge length $\ell$ is $$T={\frac {\sqrt {3}}{4}}\ell ^{2}.$$ The formula may be derived from the formula of an isosceles triangle by Pythagoras theorem: the altitude $h$ of a triangle is the square root of the difference of squares of a side and half of a base. Since the base and the legs are equal, the height is: $$h={\sqrt {\ell ^{2}-{\frac {\ell ^{2}}{4}}}}={\frac {\sqrt {3}}{2}}\ell .$$ In general, the area of a triangle is half the product of its base and height. The formula for the area of an equilateral triangle can be obtained by substituting the altitude formula.

A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, for perimeter $p$ and area $T$, the equality holds for the equilateral triangle:$$p^{2}=12{\sqrt {3}}T.$$

Symmetry

Main article: Dihedral group of order 6

Illustration of the six isometries of an equilateral triangle, preserving its symmetry

The equilateral triangle has the symmetry of a dihedral group $\mathrm {D} _{3}$ of order six: this group allows to preserve its appearance by six transformations, three reflections line passing through a vertex to an edge's midpoint by flipping it across and three rotations around its center for one-third of a full turn (0°, 120°, 240°).

Incircle and circumcircle

The inscribed circle of an equilateral triangle is a circle that is fit inside such a triangle, where it is tangent to the edge of an equilateral triangle. The formula for the inradius is the distance between the centroid of a circle (i.e., the centroid of an equilateral triangle) to any midedge of an equilateral triangle $r={\sqrt {3}}\ell /6$. The circumscribed circle of an equilateral triangle is a circle that fits an equilateral triangle, where it is tangent to all three vertices of such a triangle. The circumradius is the distance from the centroid of an equilateral triangle to any vertex of the triangle $R=\ell /{\sqrt {3}}$.

A theorem of Euler states that the distance $t$ between circumcenter and incenter is formulated as $t^{2}=R(R-2r)$. As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius $R$ to the inradius $r$ of any triangle. That is:$$R\geq 2r.$$

Pompeiu's and van Schooten's theorem illustrations

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.

Applications

Equilateral triangle usage as a yield sign

A triangular rack

Equilateral triangles have frequently appeared in man-made constructions and in popular culture. In architecture, an example can be seen in the cross-section of the Gateway Arch, the surface of the Vegreville egg, and the cathedral Sant'Ivo alla Sapienza floor plan. It appears in the flag of Nicaragua and the flag of the Philippines. It is a shape of a variety of road signs, including the yield sign. The cue sports of 8-ball pool arranged fifteen object balls with a tool of an equilateral triangular shaped rack. The chess variants, namely tri-chess and triangular chess, invented by George R. Dekle Sr. in 1986, have boards with an equilateral triangular grid..

The ternary plot of flammability diagram, showing which mixtures of methane, oxygen gas, and inert nitrogen gas burn

The ternary plot uses the equilateral triangle to depict graphically the ratios of the three variables as positions in an equilateral triangle, with an applications are in physical chemistry, petrology, mineralogy, metallurgy, and other physical sciences to show the compositions of systems composed of three species. The plot has different names such as the de Finetti diagram in the mathematical modellilng of population genetics, and a simplex plot in game theory and convex optimization.

Equilateral triangles are used in the study of science. In the study of stereochemistry, it can be described as the molecular geometry in which one atom in the center connects three other atoms in a plane, known as the trigonal planar molecular geometry. In electrocardiography, the formation of an imaginary inverted triangle by two arms and the leg with the chest as its center, and this triangle is called Einthoven's triangle, named after Dutch medical doctor and psychology Willem Einthoven. In the Thomson problem, concerning the minimum-energy configuration of $n$ charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the best solution known for $n=3$ places the points at the vertices of an equilateral triangle, inscribed in the sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.

Constructions

Extract from the 1826 Ladies' Diary

Let $\triangle ABG$, $\triangle BHC$, and $\triangle ACK$ be some equilateral triangles constructed on the sides of an arbitrary triangle $ABC$. Either all outward or inward, the centers of the three equilateral triangles $D$, $E$, and $F$, respectively form an equilateral triangle $\triangle DEF$. The result is attributed to Emperor of the French Napoleon Bonaparte by many mathematicians, thereby it is named Napoleon's theorem, although Napoleon had no connection to the mathematical works. Along with the similar event of the construction problem on circle and its center by a compass, these two were discovered by Italian geometer and mathematician Lorenzo Mascheroni, who let the Emperor claim them for himself. The Ladies' Diary published 1825 showed a proof of Napoleon's theorem.

Compass

The very first proposition in the Elements by Euclid starts by drawing a circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles intersect in two points. An equilateral triangle can be constructed by joining the two centers of the circles and one of the points of intersection.

Construction of equilateral triangle with compass and straightedge

A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinct Fermat primes. There are five known Fermat primes: 3, 5, 17, 257, 65537.

Begin with any line segment as one side; place the point of the compass on one end of the line, then swing an arc from that point to the other point of the line segment; repeat with the other side of the line, which connects the point where the two arcs intersect with each end of the line segment in the aftermath.

Related topics

A packing problem asks the objective of $n$ circles packing into the smallest possible equilateral triangle. Optimal solutions are known for packing $n<13$ circles into an equilateral triangle, and conjecturally optimal solutions for $13\leq n<28$.

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

The illustrated proof of Viviani's theorem by rotating the altitude of three inscribed equilateral triangles

Let $\triangle ABC$ be an equilateral triangle and $P$ be any point inside $\triangle ABC$. Let $\triangle DGP$, $\triangle EHP$, and $\triangle FIP$ be some equilateral triangles within $\triangle ABC$. Draw the altitude of those three triangles from the distance between $P$ to the midpoints of $AC$, $BC$, and $AB$, respectively. By rotating these segment lines vertically, the sum of $AC$, $BC$, and $AB$ is equal to the altitude of $\triangle ABC$. This result is from Viviani's theorem, named after Italian mathematician Vincenzo Viviani.

An equilateral triangle may have integer sides with three rational angles as measured in degrees, known for the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes), and the only triangle whose Steiner inellipse is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral. It is the only regular polygon aside from the square that can be inscribed inside any other regular polygon.

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. 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 $\angle APB$, $\angle BPC$, and $\angle CPA$ cross the sides ($A$, $B$, and $C$ being the vertices). There are numerous other triangle inequalities that hold equality if and only if the triangle is equilateral.

Counting and numbers

Main articles: Padovan sequence and Triangular numbers

The Padovan sequence and the spiral of equilateral triangles

The Padovan sequence is the sequence of integers with the given initial values $P_{1}=P_{2}=P_{3}=1$ and the recurrence relation $$P_{n}=P_{n-2}+P_{n-3}.$$ The resulting recurrence relation generates a sequence consisting of the first few numbers:

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, ... (sequence A000931 in the OEIS)

The Padovan sequence is named after Richard Padovan, who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay. Geometrically, the sequence is describable as the tiling of equilateral triangles based on the edge length successively, creating a spiral. The result is analogous to the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, ... with the tiling of squares.

The first six triangular numbers originated from $T_{1}=1$ instead of $T_{0}=0$

The triangular numbers are the type of figurate numbers, the sequence of positive integers that describe geometrically the arrangement of points in an equilateral triangle. Started with one point, continuing with two points, then three points, and so on. The sequence tells the total of points as it goes, and calculating the total for $n$-th term the formula gives $n(n+1)/2$. The first few numbers are:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... (sequence A000217 in the OEIS)

Triangular numbers were applications for Pythagoreanism; for example, tetractys is a triangular figure consisting of ten points arranged in four rows symbolizing the harmony of music and cosmos, the ascent of divines, the classical elements, and the generalization of triangular mathematical spaces called simplices.

Fractal, polytopes, and other shapes

The equilateral triangle tiling fills the plane

The Sierpiński triangle

The regular octahedron is a deltahedron as well as a member of the family of antiprisms

Notably, the equilateral triangle tiles the Euclidean plane with six triangles meeting at a vertex; the dual of this tessellation is the hexagonal tiling. Truncated hexagonal tiling, rhombitrihexagonal tiling, trihexagonal tiling, snub square tiling, and snub hexagonal tiling are all semi-regular tessellations constructed with equilateral triangles. Other two-dimensional objects built from equilateral triangles include the Sierpiński triangle (a fractal shape constructed from an equilateral triangle by subdividing recursively into smaller equilateral triangles) and Reuleaux triangle (a curved triangle with constant width, constructed from an equilateral triangle by rounding each of its sides).

Equilateral triangles may also form a polyhedron in three dimensions. A polyhedron whose faces are all equilateral triangles is called a deltahedron. There are eight strictly convex deltahedra: three of the five Platonic solids (regular tetrahedron, regular octahedron, and regular icosahedron) and five of the 92 Johnson solids (triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, and gyroelongated square bipyramid). More generally, all Johnson solids have equilateral triangles among their faces, though most also have other regular polygons. The antiprisms are a family of polyhedra incorporating a band of alternating triangles. When the antiprism is uniform, its bases are regular and all triangular faces are equilateral.

As a generalization, the equilateral triangle belongs to the infinite family of $n$-simplexes, with $n=2$.

See also

• Clifton Cathedral
• Dirichlet distribution
• Eternity puzzle
• Almost-equilateral Heronian triangle
• Hofstadter points
• Illusory contour
• Kingittorsuaq Runestone
• Lemoine's problem
• Malfatti circles
• Mohr–Mascheroni theorem
• Nesbitt's inequality
• Pipeclay triangle
• Sphinx tiling
• Steiner tree problem
• Thébault's theorem
• Three Pagodas
• Trilinear coordinates

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External links

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