Heronian triangle

Template:Short description In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths Template:Mvar, Template:Mvar, and Template:Mvar and area Template:Mvar are all positive integers.<ref name=carlson>Template:Citation</ref><ref>Template:Citation </ref> Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demonstrated with the example triangle of sides Template:Math and area Template:Math.<ref>Template:Cite journal</ref>

Heron's formula implies that the Heronian triangles are exactly the positive integer solutions of the Diophantine equation

<math>16\,A^2=(a+b+c)(a+b-c)(b+c-a)(c+a-b);</math>

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.<ref>The sides and area of any triangle satisfy the Diophantine equation obtained by squaring both sides of Heron's formula; see Template:Slink. Conversely, consider a solution of the equation where <math>(a, b, c, A)</math> are all positive integers. It corresponds to a valid triangle if and only if the triangle inequality is satisfied, that is, if the three integers <math>a+b-c,</math> <math>b+c-a,</math> and <math>c+a-b</math> are all positive. This is necessarily true in this case: if any of these sums were negative or zero, the other two would be positive and the right-hand side of the equation would thus be negative or zero and could not possibly equal the left-hand side <math>16\,A^2,</math> which is positive.</ref>

If the three side lengths are setwise coprime (meaning that the greatest common divisor of all three sides is 1), the Heronian triangle is called primitive.

Triangles whose side lengths and areas are all rational numbers (positive rational solutions of the above equation) are sometimes also called Heronian triangles or rational triangles;<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:HeronianTriangle%7CHeronianTriangle.html}} |title = Heronian Triangle |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> 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 geometrically similar to exactly one primitive Heronian triangle.

In any rational Heronian triangle, the three altitudes, the circumradius, the inradius and exradii, and the sines and cosines of the three angles are also all rational numbers.

Scaling to primitive triangles

Scaling a triangle with a factor of Template:Mvar consists of multiplying its side lengths by Template:Mvar; this multiplies the area by <math>s^2</math> and produces a similar triangle. Scaling a rational Heronian triangle by a rational factor produces another rational Heronian triangle.

Given a rational Heronian triangle of side lengths <math display=inline>\frac pd, \frac qd,\frac rd,</math> the scale factor <math display=inline>\frac d{\gcd(p,q,r)}</math> produces a rational Heronian triangle such that its side lengths <math display=inline>a, b,c</math> are setwise coprime integers. It is proved below that the area Template:Mvar 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 <math display=inline>a, b,c</math> of a rational Heronian triangle are coprime integers, then the area Template:Mvar is also an integer and exactly one of the side lengths is even.

The Diophantine equation given in the introduction shows immediately that <math>16A^2</math> is an integer. Its square root <math>4A</math> is also an integer, since the square root of an integer is either an integer or an irrational number.

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 Template:Math, one gets that <math>A^2</math> and <math>A</math> 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 Template:Mvar is odd, the right-hand side of the Diophantine equation can be rewritten

<math>((a+b)^2-c^2)(c^2-(a-b)^2),</math>

with <math>a+b</math> and <math>a-b</math> even. As the square of an odd integer is congruent to <math>1</math> modulo Template:Math, the right-hand side of the equation must be congruent to <math>-1</math> modulo Template:Math. It is thus impossible, that one has a solution of the Diophantine equation, since <math>16A^2</math> must be the square of an integer, and the square of an integer is congruent to Template:Math or Template:Math modulo Template:Math.

Examples

Any Pythagorean triangle is a Heronian triangle. The side lengths of such a triangle are integers, 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.

File:Triangle-heronian.svg

A triangle with sidelengths Template:Mvar, Template:Mvar and Template:Math, and height Template:Mvar.

Examples of Heronian triangles that are not right-angled are the isosceles triangle obtained by joining a Pythagorean triangle and its mirror image along a side of the right angle. Starting with the Pythagorean triple Template:Math this gives two Heronian triangles with side lengths Template:Math and Template:Math and area Template:Math.

More generally, given two Pythagorean triples <math>(a,b,c)</math> and <math>(a,d,e)</math> with largest entries Template:Mvar and Template:Mvar, one can join the corresponding triangles along the sides of length Template:Mvar (see the figure) for getting a Heronian triangle with side lengths <math>c,e,b+d</math> and area <math display=inline>\tfrac12a(b+d)</math> (this is an integer, since the area of a Pythagorean triangle is an integer).

There are Heronian triangles that cannot be obtained by joining Pythagorean triangles. For example, the Heronian triangle of side lengths <math>5, 29, 30</math> and area 72, since none of its altitudes is an integer. Such Heronian triangles are known as Template:Em.<ref name=Yiu>Template:Citation</ref> However, every Heronian triangle can be constructed from right triangles with rational side lengths, and is thus similar to a decomposable Heronian triangle. In fact, at least one of the altitudes of a triangle is inside the triangle, and divides it into two right triangles. These triangles have rational sides, since the cosine and the sine of the angles of a Heronian triangle are rational numbers, and, with notation of the figure, one has <math>a=c\sin \alpha</math> and <math>b=c\cos\alpha,</math> where <math>\alpha</math> is the left-most angle of the triangle.

Rationality properties

Many quantities related to a Heronian triangle are rational numbers. In particular:

• All the altitudes of a Heronian triangle are rational.<ref name=Somos>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> 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 <math>p_a=\tfrac{2aA}{a^2+b^2-c^2},</math> <math>p_b=\tfrac{2bA}{a^2+b^2-c^2},</math> and <math>p_c=\tfrac{2cA}{a^2-b^2+c^2},</math> where the sides are Template:Math and the area is Template:Mvar;<ref>Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", Forum Geometricorum 13, 53−59: Theorem 2.</ref> in a Heronian triangle all of Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are integers.
• Every interior angle of a Heronian triangle has a rational sine. This follows from the area formula Template:Math, in which the area and the sides Template:Mvar and Template:Mvar 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, Template:Math, in which the sides Template:Mvar, Template:Mvar, and Template:Mvar 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 Template:Math, and equivalently for other interior angles. Knowledge of (at least two 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 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 polygons generally; if all such central angles have rational tangents for their quarter angles then the cyclic polygon can be scaled to simultaneously have integer area, sides, and diagonals (connecting any two vertices).
• 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.<ref name=Zelator>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 Template:Mvar has length <math>\tfrac{2Aa}{a^2+2A}</math> where Template:Mvar is the triangle's area;<ref>Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", Mathematics Magazine 71(4), 1998, 278–284.</ref> in a Heronian triangle, both Template:Mvar and Template:Mvar are integers.
• Every Heronian triangle has a rational inradius (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 (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 to each side is rational because, for all triangles, this distance is the ratio of twice the area to three times the side length.<ref>Clark Kimberling, "Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers", Forum Geometricorum, 10 (2010), 135−139. http://forumgeom.fau.edu/FG2010volume10/FG201015index.html</ref> 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, orthocenter, nine-point center, symmedian point, Gergonne point and Nagel point.<ref name=ck>Clark Kimberling's Encyclopedia of Triangle Centers {{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

• Every Heronian triangle can be placed on a unit-sided square lattice with each vertex at a lattice point.<ref name=Yiu3>Template:Cite journal</ref> As a corollary, every rational Heronian triangle can be placed into a two-dimensional Cartesian coordinate system with all rational-valued coordinates.

Properties of side lengths

Here are some properties of side lengths of Heronian triangles, whose side lengths are Template:Math and area is Template:Mvar.

• Every primitive Heronian triangle has one even and two odd sides (see Template:Slink). It follows that a Heronian triangle has either one or three sides of even length,<ref name=Buchholz1>Template:Cite journal</ref>Template:Rp and that the perimeter of a primitive Heronian triangle is always an even number.<ref name=Friche>Template:Cite arXiv</ref>
• There are no equilateral Heronian triangles, since a primitive Heronian triangle has one even side length and two odd side lengths.<ref name=Somos/>
• The area of a Heronian triangle is always divisible by 6.<ref>Proof. One can suppose that the Heronian triangle is primitive. The right-hand side of the Diophantine equation can be rewritten as <math>((a+b)^2-c^2)(c^2-(a-b)^2).</math> If an odd length is chosen for Template:Mvar, all squares are odd, and therefore of the form <math>8k+1;</math> and the two differences are multiple of Template:Math. So <math>16A^2</math> is multiple of Template:Math, and Template:Mvar is even. For the divisibility by three, one chooses Template:Mvar as non-multiple of Template:Math (the triangle is supposed to be primitive). If one of <math>a+b</math> and <math>a-b</math> is not a multiple of Template:Mvar, the corresponding factor is a nultiple of Template:Mvar (since the square of a non-multiple of Template:Mvar has the form <math>3k+1</math>), and this implies that Template:Mvar is a divisor of <math>16A^2.</math> Otherwise, Template:Mvar would divide both <math>a+b</math> and <math>a-b,</math> and the right-hand side of the Diophantine would not be the square of <math>4A,</math> as being congruent to minus times a square modulo Template:Math. So this last case is impossible.</ref><ref name=Friche/>
• There are no Heronian triangles with a side length of either 1 or 2.<ref>Proof. Supposing <math>a\ge b\ge c,</math> the triangle inequality implies <math>b\le a\le b+c.</math> If <math>c=1,</math> this implies that <math>a=b,</math> and the condition that there is exactly one even side length cannot be fulfilled. If <math>c=2,</math> one has two even side lengths if <math>a=b+1.</math> So, <math>a=b,</math> and the Diophantine equation becomes <math>16 A^2=16(a^2-1),</math> which is impossible for two positive integers.</ref><ref name=carlson/>
• There exist an infinite number of primitive Heronian triangles with one side length equal to a given Template:Mvar, provided that Template:Math.<ref name=carlson/>
• The semiperimeter Template:Mvar of a Heronian triangle cannot be prime (as <math>s(s-a)(s-b)(s-c)</math> is the square of the area, and the area is an integer, if Template:Mvar were prime, it would divide another factor; this is impossible as these factors are all less than Template:Mvar).
• In a Heronian triangles that has no integer altitude (indecomposable and non-Pythagorean), all side lengths have a prime factor of the form Template:Math.<ref name=Yiu/> In a primitive Pythagoran triangle, all prime factors of the hypotenuse have the form Template:Math. A decomposable Heronian triangle must have two sides that are the hypotenuse of a Pythagorean triangle, and thus two sides that have prime factors of the form Template:Math. There may also be prime factors of the form Template:Math, since the Pythagorean components of a decomposable Heronian triangle need not to be primitive, even if the Heronian triangle is primitive. In summary, all Heronian triangles have at least one side that is divisible by a prime of the form Template:Math.
• There are no Heronian triangles whose side lengths form a geometric progression.<ref name=Buchholz>Template:Cite journal</ref>
• If any two sides (but not three) of a Heronian triangle have a common factor, that factor must be the sum of two squares.<ref>Template:Cite journal</ref>

Parametrizations

A parametric equation or parametrization of Heronian triangles consists of an expression of the side lengths and area of a triangle as functionsTemplate:Mdashtypically polynomial functionsTemplate:Mdashof some parameters, such that the triangle is Heronian if and only if the parameters satisfy some constraintsTemplate:Mdashtypically, 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 (598-668 A.D.), who did not prove that all Heronian triangles can be generated by the parametrization. In the 18th century, Leonhard Euler 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 parametrizationTemplate:Mdashthat is a parametrization where the parameters are positive rational numbersTemplate:Mdashis naturally derived from properties of Heronian triangles. Both Brahmagupta's and Euler's parametrizations can be recovered from this rational parametrization by clearing denominators. This provides a proof that Brahmagupta's and Euler's parametrizations generate all Heronian triangles.

Brahmagupta's parametric equation

The Indian mathematician Brahmagupta (598-668 A.D.) discovered the following parametric equations for generating Heronian triangles,<ref name="Kurz">Template:Cite journal.</ref> but did not prove that every similarity class of Heronian triangles can be obtained this way.Template:Citation needed

For three positive integers Template:Mvar, Template:Mvar and Template:Mvar that are setwise coprime (<math>\gcd(m,n,k)=1</math>) and satisfy <math>mn > k^2</math> (to guarantee positive side lengths) and Template:Nobr (for uniqueness):

<math>\begin{align}

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^{2}), & r &= k(mn - k^2), \\ \end{align}</math>

where Template:Mvar is the semiperimeter, Template:Mvar is the area, and Template:Mvar 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 Template:Math, Template:Math and Template:Math produces a triangle with Template:Math, Template:Math and Template:Math, which is similar to the Template:Math Heronian triangle with a proportionality factor of Template:Math.

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 <math>\gcd(a,b,c)</math> cannot be predicted.<ref name="Kurz" />

Euler's parametric equation

The following method of generating all Heronian triangles was discovered by Leonhard Euler,Template:Sfn who was the first to provably parametrize all such triangles.

For four positive integers Template:Mvar coprime to Template:Mvar and Template:Mvar coprime to Template:Mvar Template:Nobr satisfying <math>mp > nq</math> (to guarantee positive side lengths):

<math>\begin{align}

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{align}</math>

where Template:Mvar is the semiperimeter, Template:Mvar is the area, and Template:Mvar is the inradius.

Even when Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are pairwise coprime, the resulting Heronian triangle may not be primitive. In particular, if Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are all odd, the three side lengths are even. It is also possible that Template:Mvar, Template:Mvar, and Template:Mvar have a common divisor other than Template:Math. For example, with Template:Math, Template:Math, Template:Math, and Template:Math, one gets Template:Math, where each side length is a multiple of Template:Math; the corresponding primitive triple is Template:Math, which can also be obtained by dividing the triple resulting from Template:Math by two, then exchanging Template:Mvar and Template:Mvar.

Half-angle tangent parametrization

File:Triangle-tikz.svg

A triangle with side lengths and interior angles labeled as in the text

Let <math>a, b, c > 0</math> be the side lengths of any triangle, let <math>\alpha, \beta, \gamma</math> be the interior angles opposite these sides, and let <math display=inline>t = \tan\frac\alpha2,</math> <math display=inline>u = \tan\frac\beta2,</math> and <math display = inline>v = \tan\frac\gamma2</math> be the half-angle tangents. The values <math>t, u, v</math> are all positive and satisfy <math>tu + uv + vt = 1</math>; this "triple tangent identity" is the half-angle tangent version of the fundamental triangle identity written as <math display=inline>\frac\alpha 2 + \frac\beta 2 + \frac\gamma 2 = \frac\pi 2</math> radians (that is, 90°), as can be proved using the addition formula for tangents. By the laws of sines and cosines, all of the sines and the cosines of <math>\alpha, \beta, \gamma</math> are rational numbers if the triangle is a rational Heronian triangle and, because a half-angle tangent is a rational function of the sine and cosine, it follows that the half-angle tangents are also rational.

Conversely, if <math>t, u, v</math> are positive rational numbers such that <math>tu + uv + vt = 1,</math> it can be seen that they are the half-angle tangents of the interior angles of a class of similar Heronian triangles.<ref>Template:Cite journal</ref> The condition <math>tu + uv + vt = 1</math> can be rearranged to <math display=inline>v = \frac{1-tu}{t+u},</math> and the restriction <math>v > 0</math> requires <math>tu < 1.</math> Thus there is a bijection between the similarity classes of rational Heronian triangles and the pairs of positive rational numbers <math>(t, u)</math> whose product is less than Template:Math.

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:<ref>Template:Cite journal</ref>

<math>\begin{align}

a &= \sin\alpha = \frac{2t}{1+t^2}, & s - a = \frac{2u(1-tu)}{(1+t^2)(1+u^2)}, \\[5mu] b &= \sin\beta = \frac{2u}{1+u^2}, & s - b = \frac{2t(1-tu)}{(1+t^2)(1+u^2)}, \\[5mu] c &= \sin\gamma = \frac{2(t+u)(1-tu)}{(1+t^2)(1+u^2)},

                                       & s - c = \frac{2tu(t+u)}{(1+t^2)(1+u^2)}, \\[5mu]

& & s = \frac{2(t+u)}{(1+t^2)(1+u^2)}, \\ A &= \frac{4tu(t+u)(1-tu)}{(1+t^2)^2(1+u^2)^2}, & r = \frac{2tu(1-tu)}{(1+t^2)(1+u^2)}, \end{align}</math> where <math>s = \tfrac12(a + b + c)</math> is the semiperimeter, <math>A = \tfrac12 ab \sin \gamma</math> is the area, <math>r = \sqrt{\tfrac{(s-a)(s-b)(s-c)}{s}}</math> is the inradius, and all these values are rational because <math>t</math> and <math>u</math> are rational.

To obtain an (integral) Heronian triangle, the denominators of Template:Mvar, Template:Mvar, and Template:Mvar must be cleared. There are several ways to do this. If <math>t = m/n</math> and <math>u = p/q,</math> with <math>\gcd(m, n) = \gcd(p,q) = 1</math> (irreducible fractions), and the triangle is scaled up by <math>\tfrac12(m^2 + n^2)(p^2 + q^2),</math> the result is Euler's parametrization. If <math>t = m/k</math> and <math>u = n/k</math> with <math>\gcd(m, n, k) = 1</math> (lowest common denominator), and the triangle is scaled up by <math>(k^2 + m^2)(k^2 + n^2)/2k,</math> the result is similar but not quite identical to Brahmagupta's parametrization. If, instead, this is <math>1/t</math> and <math>1/u</math> that are reduced to the lowest common denominator, that is, if <math>t = k/m</math> and <math>u = k/n</math> with <math>\gcd(m, n, k) = 1,</math> then one gets exactly Brahmagupta's parametrization by scaling up the triangle by <math>(k^2 + m^2)(k^2 + n^2)/2k.</math>

This proves that either parametrization generates all Heronian triangles.

The values of Template:Mvar, Template:Mvar and Template:Mvar that give the set of triangles that are geometrically similar to the triangle with side lengths Template:Mvar, Template:Mvar, and Template:Mvar, semiperimeter Template:Tmath, and area Template:Mvar are <math display=block>(t, u, v) = \left( \frac{A}{s(s-a)}, \frac{A}{s(s-b)}, \frac{A}{s(s-c)} \right)\,.</math>

Other results

Template:Harvtxt 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 <math>r</math> and all three of the exradii <math>(r_a, r_b, r_c)</math>, including the ones generated by<ref name=Yiu1>Zhou, Li, "Primitive Heronian Triangles With Integer Inradius and Exradii", Forum Geometricorum 18, 2018, 71-77. http://forumgeom.fau.edu/FG2018volume18/FG201811.pdf</ref>Template:Rp

<math>\begin{align}

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{align}</math>

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.<ref name=Yiu1/>Template:Rp

See also Template:Slink 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, starts as in the following table. "Primitive" means that the greatest common divisor of the three side lengths equals 1.

Area	Perimeter	side length Template:Math	side length Template:Mvar	side length Template:Mvar
6	12	5	4	3
12	16	6	5	5
12	18	8	5	5
24	32	15	13	4
30	30	13	12	5
36	36	17	10	9
36	54	26	25	3
42	42	20	15	7
60	36	13	13	10
60	40	17	15	8
60	50	24	13	13
60	60	29	25	6
66	44	20	13	11
72	64	30	29	5
84	42	15	14	13
84	48	21	17	10
84	56	25	24	7
84	72	35	29	8
90	54	25	17	12
90	108	53	51	4
114	76	37	20	19
120	50	17	17	16
120	64	30	17	17
120	80	39	25	16
126	54	21	20	13
126	84	41	28	15
126	108	52	51	5
132	66	30	25	11
156	78	37	26	15
156	104	51	40	13
168	64	25	25	14
168	84	39	35	10
168	98	48	25	25
180	80	37	30	13
180	90	41	40	9
198	132	65	55	12
204	68	26	25	17
210	70	29	21	20
210	70	28	25	17
210	84	39	28	17
210	84	37	35	12
210	140	68	65	7
210	300	149	148	3
216	162	80	73	9
234	108	52	41	15
240	90	40	37	13
252	84	35	34	15
252	98	45	40	13
252	144	70	65	9
264	96	44	37	15
264	132	65	34	33
270	108	52	29	27
288	162	80	65	17
300	150	74	51	25
300	250	123	122	5
306	108	51	37	20
330	100	44	39	17
330	110	52	33	25
330	132	61	60	11
330	220	109	100	11
336	98	41	40	17
336	112	53	35	24
336	128	61	52	15
336	392	195	193	4
360	90	36	29	25
360	100	41	41	18
360	162	80	41	41
390	156	75	68	13
396	176	87	55	34
396	198	97	90	11
396	242	120	109	13

The list of primitive Heronian triangles whose sides do not exceed 6,000,000 has been computed by Template:Harvtxt.

Heronian triangles with perfect square sides

As of February 2021, only two primitive Heronian triangles with perfect square sides are known:

(1853², 4380², 4427², Area=Template:Val), published in 2013.<ref>Template:Cite journal</ref>

(11789², 68104², 68595², Area=Template:Val), published in 2018.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Heronian triangles with perfect square sides are connected to the Perfect cuboid problem. The existence of a solution to the Perfect cuboid problem is equivalent to the existence of a solution to the Perfect square triangle problem:<ref>Florian Luca, Perfect Cuboids and Perfect Square Triangles (2000) Mathematics Magazine Vol. 73, No. 5 (Dec., 2000), pp. 400–401</ref> "Does there exist a triangle whose side lengths are perfect squares and whose angle bisectors are integers?".

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),Template:Sfn<ref>Template:Citation</ref> though only four of them are primitive.

Almost-equilateral Heronian triangles

Since the area of an equilateral triangle with rational sides is an irrational number, 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 (sequence A102341 in the OEIS):

Side length			Area
Template:Mvar	Template:Math	Template:Mvar
5	5	6	12
17	17	16	120
65	65	66	Template:Val
241	241	240	Template:Val
901	901	902	Template:Val
Template:Val	Template:Val	Template:Val	Template:Val
Template:Val	Template:Val	Template:Val	Template:Val
Template:Val	Template:Val	Template:Val	Template:Val

There is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form Template:Math, Template:Mvar, Template:Math. A method for generating all solutions to this problem based on continued fractions was described in 1864 by Edward Sang,<ref>Template:Citation. See in particular p. 734.</ref> and in 1880 Reinhold Hoppe gave a closed-form expression for the solutions.<ref>Template:Citation.</ref> The first few examples of these almost-equilateral triangles are listed in the following table (sequence A003500 in the OEIS):

Side length			Area	Inradius
Template:Math	Template:Mvar	Template:Math
1	2	3	0	0
3	4	5	6	1
13	14	15	84	4
51	52	53	Template:Val	15
193	194	195	Template:Val	56
723	724	725	Template:Val	209
Template:Val	Template:Val	Template:Val	Template:Val	780
Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
Template:Val	Template:Val	Template:Val	Template:Val	Template:Val

Subsequent values of Template:Mvar can be found by multiplying the previous value by 4, then subtracting the value prior to that one (Template:Nowrap, Template:Nowrap, etc.), thus:

<math>n_t = 4n_{t-1} - n_{t-2} \, ,</math>

where Template:Mvar denotes any row in the table. This is a Lucas sequence. Alternatively, the formula <math>(2 + \sqrt{3})^t + (2 - \sqrt{3})^t</math> generates all Template:Mvar for positive integers Template:Mvar. Equivalently, let Template:Math and Template:Math, then,

<math>\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</math>

where Template:Math are solutions to Template:Math. A small transformation Template:Math yields a conventional Pell equation Template:Math, the solutions of which can then be derived from the regular continued fraction expansion for Template:Radic.<ref>Template:Citation</ref>

The variable Template:Mvar is of the form <math>n=\sqrt{2 + 2 k}</math>, where Template:Mvar is 7, 97, 1351, 18817, .... The numbers in this sequence have the property that Template:Mvar consecutive integers have integral standard deviation.<ref>Online Encyclopedia of Integer Sequences, Template:OEIS2C.</ref>

See also

• Heronian tetrahedron
• Brahmagupta quadrilateral
• Brahmagupta triangle
• Robbins pentagon
• Integer triangle#Heronian triangles

Further reading

• Template:Cite book
• Template:Cite book

External links

• Template:Mathworld
• Online Encyclopedia of Integer Sequences Heronian
• Template:Citation