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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Heronian triangle (or Heron triangle) is a
triangle 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 \triangle ABC. In Euclidean geometry, any three points, when non-colline ...
whose side lengths , , and and
area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ...
are all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. Heronian triangles are named after
Heron of Alexandria Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He i ...
, based on their relation to
Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-centur ...
. Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
:16\,A^2=(a+b+c)(a+b-c)(b+c-a)(c+a-b); that is, the side lengths and area of any Heronian triangle satisfy the equation, and any positive integer solution of the equation describes a Heronian triangle. If the three side lengths are setwise coprime, the Heronian triangle is called ''primitive''. Triangles whose side lengths and areas are all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s (positive rational solutions of the above equation) are sometimes also called ''Heronian triangles'' or rational triangles; in this article, these more general triangles will be called ''rational Heronian triangles''. Every (integral) Heronian triangle is a rational Heronian triangle. Conversely, every rational Heronian triangle is similar to exactly one primitive Heronian triangle.


Scaling to primitive triangles

Scaling Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
a triangle with a factor of consists of multiplying its side lengths by ; this multiplies the area by s^2 and produces a similar triangle. Scaling a rational Heronian triangle by a
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
factor produces another rational Heronian triangle. Given a rational Heronian triangle of side lengths \frac pd, \frac qd,\frac rd, the scale factor \frac d produce a rational Heronian triangle such that its side lengths a, b,c are setwise coprime numbers. It is proved below that the area is an integer, and thus the triangle is a Heronian triangle. Such a triangle is often called a ''primitive Heronian triangle.'' In summary, every similarity class of rational Heronian triangles contains exactly one primitive Heronian triangle. A byproduct of the proof is that exactly one of the side lengths of a primitive Heronian triangle is an even integer. ''Proof:'' One has to prove that, if the side lengths a, b,c of a rational Heronian triangle are coprime integers, then the area is also an integer and exactly one of the side lengths is even. The Diophantine equation given in the introduction shows immediately that 16A^2 is an integer. Its square root 4A is also an integer, since the square root of an integer is either an integer or an
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
. If exactly one of the side lengths is even, all the factors in the right-hand side of the equation are even, and, by dividing the equation by , one gets that A^2 and A are integers. As the side lengths are supposed to be coprime, one is left with the case where one or three side lengths are odd. Supposing that is odd, the right-hand side of the Diophantine equation can be rewritten :((a+b)^2-c^2)(c^2-(a-b)^2), with a+b and a-b even. As the square of an odd integer is congruent to 1 modulo , the right-hand side of the equation must be congruent to -1 modulo . It is thus impossible, that one has a solution of the Diophantine equation, since 16A^2 must be the square of an integer, and the square of an integer is congruent to or modulo .


Examples

Any
Pythagorean triangle A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
is a Heronian triangle. The side lengths of such a triangle are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, by definition. In any such triangle, one of the two shorter sides has even length, so the area (the product of these two sides, divided by two) is also an integer. Examples of Heronian triangles that are not right-angled are the
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 versio ...
obtained by joining a Pythagorean triangle and its mirror image along a side of the right angle. Starting with the
Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
this gives two Heronian triangles with side lengths and and area . More generally, given two Pythagorean triples (a,b,c) and (a,d,e) with largest entries and , one can join the corresponding triangles along the sides of length (see the figure) for getting a Heronian triangle with side lengths c,e,b+d and area \tfrac12a(b+d) (this is an integer, since the area of a Pythagorean triangle is an integer). There are Heronian that cannot be obtained by joining Pythagorean triangles. For example, the Heronian triangle of side lengths 5, 29, 30 and area 72, since none of its altitudes is an integer. Such Heronian triangles are known as . However, every Heronian triangle can be constructed from right triangles with
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
side lengths, since every altitude of a Heronian triangle is a rational number, the ratio of twice the area and a side length.


Rationality properties

Many quantities related to a Heronian triangle are rational numbers. In particular: *All the altitudes of a Heronian triangle are rational. This can be seen from the fact that the area of a triangle is half of one side times its altitude from that side, and a Heronian triangle has integer sides and area. Some Heronian triangles have three non-integer altitudes, for example the acute (15, 34, 35) with area 252 and the obtuse (5, 29, 30) with area 72. Any Heronian triangle with one or more non-integer altitudes can be scaled up by a factor equalling the least common multiple of the altitudes' denominators in order to obtain a similar Heronian triangle with three integer altitudes. *All the interior perpendicular bisectors of a Heronian triangle are rational: For any triangle these are given by p_a=\tfrac, p_b=\tfrac, and p_c=\tfrac, where the sides are ''a'' ≥ ''b'' ≥ ''c'' and the area is ''A''; in a Heronian triangle all of ''a'', ''b'', ''c'', and ''A'' are integers. *Every
interior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...
of a Heronian triangle has a rational sine. This follows from the area formula , in which the area and the sides ''a'' and ''b'' are integers, and equivalently for the other interior angles. *Every interior angle of a Heronian triangle has a rational cosine. This follows from the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
, , in which the sides ''a'', ''b'', and ''c'' are integers, and equivalently for the other interior angles. *Because all Heronian triangles have all interior angles' sines and cosines rational, this implies that the tangent, cotangent, secant, and cosecant of each interior angle is either rational or infinite. *Half of each interior angle has a rational tangent because , and equivalently for other interior angles. Knowledge of these half-angle tangent values is sufficient to reconstruct the side lengths of a primitive Heronian triangle ( see below). *For any triangle, the angle spanned by a side as viewed from the center of the
circumcircle 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. Not every poly ...
is twice the interior angle of the triangle vertex opposite the side. Because the half-angle tangent for each interior angle of a Heronian triangle is rational, it follows that the quarter-angle tangent of each such central angle of a Heronian triangle is rational. (Also, the quarter-angle tangents are rational for the central angles of a Brahmagupta quadrilateral, but is an unsolved problem whether this is true for all Robbins pentagons.) The reverse is true for all
cyclic polygon 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. Not every polyg ...
s generally; if all such central angles have rational tangents for their quarter angles then the cyclic polygon can be scaled to simultaneously have integer side lengths and integer area. *There are no Heronian triangles whose three internal angles form an arithmetic progression. This is because all plane triangles with interior angles in an arithmetic progression must have one interior angle of 60°, which does not have a rational sine.2+3y2=z2", ''Cornell Univ. archive'', 2008">Zelator, K., "Triangle Angles and Sides in Progression and the diophantine equation x2+3y2=z2", ''Cornell Univ. archive'', 2008
/ref> *Any square inscribed in a Heronian triangle has rational sides: For a general triangle the inscribed square on side of length ''a'' has length \tfrac where ''A'' is the triangle's area; in a Heronian triangle, both ''A'' and ''a'' are integers. *Every Heronian triangle has a rational
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
(radius of its inscribed circle): For a general triangle the inradius is the ratio of the area to half the perimeter, and both of these are rational in a Heronian triangle. *Every Heronian triangle has a rational
circumradius 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. Not every polyg ...
(the radius of its circumscribed circle): For a general triangle the circumradius equals one-fourth the product of the sides divided by the area; in a Heronian triangle the sides and area are integers. *In a Heronian triangle the distance from the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ob ...
to each side is rational because, for all triangles, this distance is the ratio of twice the area to three times the side length. This can be generalized by stating that all centers associated with Heronian triangles whose barycentric coordinates are rational ratios have a rational distance to each side. These centers include the
circumcenter 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. Not every poly ...
,
orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ' ...
,
nine-point center In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so called because it is the center of the nine-point circle, a circl ...
,
symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over th ...
,
Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's ince ...
and
Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concurr ...
.Clark Kimberling's Encyclopedia of Triangle Centers *Every Heronian triangle can be placed on a unit-sided
square lattice In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
with each vertex at a lattice point.Yiu, P., "Heronian triangles are lattice triangles", ''American Mathematical Monthly'' 108 (2001), 261–263. As a corollary, every rational Heronian triangle can be placed into a two-dimensional
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
with all rational-valued coordinates.


Properties of side lengths

Here are some properties of side lengths of Heronian triangles, whose side lengths are and area is . *Every primitive Heronian triangle Heronian triangle has one even and two odd sides (see ). It follows that a Heronian triangle has either one or three sides of even length, and that the perimeter of a primitive Heronian triangle is always an even number. *There are no equilateral Heronian triangles, since a primitive Heronian triangle has one even side length and two odd side lengths. *The area of a Heronian triangle is always divisible by 6. *There are no Heronian triangles with a side length of either 1 or 2. *There exist an infinite number of primitive Heronian triangles with one side length equal to a given , provided that . *The semiperimeter of a Heronian triangle cannot be prime (as s(s-a)(s-b)(s-c) is the square of the area, and the area is an integer, if would be prime, it should divide another factor; this is impossible as these factors are all less than ). *Heronian triangles that have no integer altitude ( indecomposable and non-Pythagorean) have sides that are all divisible by primes of the form . However decomposable Heronian triangles must have two sides that are the hypotenuse of Pythagorean triangles. Hence all Heronian triangles that are not Pythagorean have at least two sides that are divisible by primes of the form . All that remains are Pythagorean triangles. Therefore, all Heronian triangles have at least one side that is divisible by primes of the form . Finally if a Heronian triangle has only one side divisible by primes of the form it has to be Pythagorean with the side as the hypotenuse and the hypotenuse must be divisible by 5. *There are no Heronian triangles whose side lengths form a
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
. *If any two sides (but not three) of a Heronian triangle have a common factor, that factor must be the sum of two squares.


Parametrizations

A
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
or ''parametrization'' of Heronian triangles consists of an expression of the side lengths and area of a triangle as functionstypically
polynomial function In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
sof some parameters, such that the triangle is Heronian if and only if the parameters satisfy some constraintstypically, to be positive integers satisfying some inequalities. It is also generally required that all Heronian triangles can be obtained up to a scaling for some values of the parameters, and that these values are unique, if an order on the sides of the triangle is specified. The first such parametrization was discovered by
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
(598-668 A.D.), who did not prove that all Heronian triangles can be generated by the parametrization. In the 18th century,
Leonard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries i ...
provided another parametrization and proved that it generates all Heronian triangles. These parametrizations are described in the next two subsections. In the third subsection, a rational parametrizationthat is a parametrization where the parameters are positive
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
is naturally derived from properties of Heronian triangles. Both Brahmagupta's and Euler's parametrizations can be recovered from this rational parametrization by
clearing denominators In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. Example Co ...
. This provides a proof that Brahmagupta's and Euler's parametrizations generate all Heronian triangles.


Brahmagupta's parametric equation

The Indian mathematician
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...
(598-668 A.D.) discovered the following
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
s for generating Heronian triangles, but did not prove that every similarity class of Heronian triangles can be obtained this way.. For three positive integers , and that are setwise coprime (\gcd(m,n,k)=1) and satisfy mn > k^2 (to guarantee positive side lengths) and (for uniqueness): :\begin a &= n(m^2 + k^2), & s - a &= \tfrac12(b + c - a) = n(mn - k^2), \\ b &= m(n^2 + k^2), & s - b &= \tfrac12(c + a - b) = m(mn - k^2), \\ c &= (m + n)(mn - k^2), & s - c &= \tfrac12(a + b - c) = (m + n)k^2, \\ && s &= \tfrac12(a + b + c) = mn(m + n), \\ A &= mnk(m+n)(mn-k^), \\ r &= k(mn - k^2), \\ \end where is the semiperimeter, is the area, and is the inradius. The resulting Heronian triangle is not always primitive, and a scaling may be needed for getting the corresponding primitive triangle. For example, taking , and produces a triangle with , and , which is similar to the Heronian triangle with a proportionality factor of . The fact that the generated triangle is not primitive is an obstacle for using this parametrization for generating all Heronian triangles with size lengths less than a given bound (since the size of \gcd(a,b,c) cannot be predicted.


Euler's parametric equation

The following method of generating all Heronian triangles was discovered by
Leonard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries i ...
, who was the first to provably parametrize all such triangles. For four positive integers coprime to and coprime to satisfying mp > nq (to guarantee positive side lengths): :\begin a &= mn(p^2 + q^2), & s - a &= mq(mp - nq), \\ b &= pq(m^2 + n^2), & s - b &= np(mp - nq), \\ c &= (mq + np)(mp - nq), & s - c &= nq(mq + np), \\ & & s &= mp(mq + np), \\ A &= mnpq(mq + np)(mp - nq), \\ r &= nq(mp - nq), \\ \end where is the semiperimeter, is the area, and is the inradius. Even when , , , and are pairwise coprime, the resulting Heronian triangle may not be primitive. In particular, if , , , and are all odd, the three side lengths are even. It is also possible that , , and have a common divisor other than . For example, with , , , and , one gets , where each side length is a multiple of ; the corresponding primitive triple is , which can also be obtained by dividing the triple resulting from by two, then exchanging and .


Half-angle tangent parametrization

Let a, b, c be the side lengths of a triangle, and \alpha, \beta, \gamma the correponding angles. By the laws of sines and cosines, all of the sines and the cosines of these angles are rational numbers if the triangle is a rational Heronian triangle. It follows that the half-angle tangents t = \tan\frac\alpha2, u = \tan\frac\beta2, and v = \tan\frac\gamma2 are also rational. Moreover, t, u, v are all positive and satisfy tu + uv + vt = 1 (this "triple tangent identity" is the half-angle tangent version of the fundamental triangle identity written as \frac\alpha 2 + \frac\beta 2+ \frac\gamma 2= \frac\pi 2, as can be proven using the addition formula for tangents). Conversely, if t, u, v are positive rational numbers such that tu + uv + vt = 1, they are the half-angle tangents of the interior angles of a class of similar Heronian triangles. The condition tu + uv + vt = 1 can be rearranged to v = \frac and v > 0 implies tu < 1. Thus there is a bijection between the similarity classes of rational Heronian triangles an the pairs of positive rational numbers whose product is less than . To make this bijection explicit, one can choose, as a specific member of the similarity class, the triangle inscribed in a unit-diameter circle with side lengths equal to the sines of the opposite angles: :\begin a &= \sin\alpha = \frac, & s - a = \frac, \\ mub &= \sin\beta = \frac, & s - b = \frac, \\ muc &= \sin\gamma = \frac, & s - c = \frac, \\ mu& & s = \frac, \\ A &= \frac, \\ mur &= \frac, \end where s = \tfrac12(a + b + c) is the semiperimeter, is the area, and is the inradius. To obtain an (integral) Heronian triangle, the denominators of , , and must be cleared. There are several ways to do this. If t = m/n and u = p/q, with \gcd(m, n) = \gcd(p,q) = 1 (
irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). ...
s), and the triangle is scaled up by \tfrac12(m^2 + n^2)(p^2 + q^2), the result is Euler's parametrization. If t = m/k and u = n/k with \gcd(m, n, k) = 1 (lowest common denomimator), and the triangle is scaled up by (k^2 + m^2)(k^2 + n^2)/2k, the result is similar but not quite identical to Brahmagupta's parametrization. If, instead, this is 1/t and 1/u that are reduced to the lowest common denominator, that is, if t = k/m and u = k/n with \gcd(m, n, k) = 1, then one gets exactly Brahmagupta's parametrization by scaling up the triangle by (k^2 + m^2)(k^2 + n^2)/2k. This proves that either parametrization generates all Heronian triangles.


Other results

has derived fast algorithms for generating Heronian triangles. There are infinitely many primitive and indecomposable non-Pythagorean Heronian triangles with integer values for the
inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
r and all three of the exradii (r_a, r_b, r_c), including the ones generated byZhou, Li, "Primitive Heronian Triangles With Integer Inradius and Exradii", ''Forum Geometricorum'' 18, 2018, 71-77. http://forumgeom.fau.edu/FG2018volume18/FG201811.pdf :\begin a &= 5(5n^2 + n - 1), & r_a &= 5n+3, \\ b &= (5n + 3)(5n^2 - 4n + 1), & r_b &= 5n^2+n-1, \\ c &= (5n - 2)(5n^2 + 6n + 2), & r_c &= (5n - 2)(5n + 3)(5n^2 + n - 1), \\ & & r &= 5n - 2, \\ A &= (5n - 2)(5n + 3)(5n^2 + n - 1) = r_c. \end There are infinitely many Heronian triangles that can be placed on a lattice such that not only are the vertices at lattice points, as holds for all Heronian triangles, but additionally the centers of the incircle and excircles are at lattice points. See also for parametrizations of some types of Heronian triangles.


Examples

The list of primitive integer Heronian triangles, sorted by area and, if this is the same, by
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several prac ...
, starts as in the following table. "Primitive" means that the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
of the three side lengths equals 1. Lists of primitive Heronian triangles whose sides do not exceed 6,000,000 can be found at


Heronian triangles with perfect square sides

Heronian triangles with perfect square sides are related to the Perfect cuboid problem. As of February 2021, only two ''primitive'' Heronian triangles with perfect square sides are known: (1853², 4380², 4427², Area=32918611718880), published in 2013. (11789², 68104² , 68595², Area=284239560530875680), published in 2018.


Equable triangles

A shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17), though only four of them are primitive.


Almost-equilateral Heronian triangles

Since the area of an
equilateral triangle 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 oth ...
with rational sides is an
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, no equilateral triangle is Heronian. However, a sequence of isosceles Heronian triangles that are "almost equilateral" can be developed from the duplication of right-angled triangles, in which the hypotenuse is almost twice as long as one of the legs. The first few examples of these almost-equilateral triangles are listed in the following table : There is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form ''n'' − 1, ''n'', ''n'' + 1. A method for generating all solutions to this problem based on
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s was described in 1864 by
Edward Sang Edward Sang FRSE FRSSA LLD (30 January 1805 – 23 December 1890) was a Scottish mathematician and civil engineer, best known for having computed large tables of logarithms, with the help of two of his daughters. These tables went beyond the ta ...
, and in 1880 Reinhold Hoppe gave a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
for the solutions. The first few examples of these almost-equilateral triangles are listed in the following table : Subsequent values of ''n'' can be found by multiplying the previous value by 4, then subtracting the value prior to that one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc.), thus: :n_t = 4n_ - n_ \, , where ''t'' denotes any row in the table. This is a
Lucas sequence In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recu ...
. Alternatively, the formula (2 + \sqrt)^t + (2 - \sqrt)^t generates all ''n'' for positive integers ''t''. Equivalently, let ''A'' = area and ''y'' = inradius, then, :\big((n-1)^2+n^2+(n+1)^2\big)^2-2\big((n-1)^4+n^4+(n+1)^4\big) = (6n y)^2 = (4A)^2 where are solutions to ''n''2 − 12''y''2 = 4. A small transformation ''n'' = ''2x'' yields a conventional
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates ...
''x''2 − 3''y''2 = 1, the solutions of which can then be derived from the regular continued fraction expansion for . The variable ''n'' is of the form n=\sqrt, where ''k'' is 7, 97, 1351, 18817, …. The numbers in this sequence have the property that ''k'' consecutive integers have integral
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
.Online Encyclopedia of Integer Sequences, .


See also

*
Heronian tetrahedron A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged ...
* Brahmagupta quadrilateral * Robbins pentagon * Integer triangle#Heronian triangles


References


Further Reading

* *


External Links

* * Online Encyclopedia of Integer Sequence
Heronian
* {{DEFAULTSORT:Heronian Triangle Arithmetic problems of plane geometry Types of triangles Articles containing proofs