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An integer triangle or integral triangle is a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
all of whose sides have lengths that 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 language ...
s. A rational triangle can be defined as one having all sides 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 abi ...
length; any such rational triangle can be integrally rescaled (can have all sides multiplied by the same integer, namely a
common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
of their denominators) to obtain an integer triangle, so there is no substantive difference between integer triangles and rational triangles in this sense. However, other definitions of the term "rational triangle" also exist: In 1914 Carmichael used the term in the sense that we today use the term
Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...
; SomosSomos, M., "Rational triangles", http://grail.eecs.csuohio.edu/~somos/rattri.html uses it to refer to triangles whose ratios of sides are rational; Conway and GuyConway, J. H., and Guy, R. K., "The only rational triangle", in ''The Book of Numbers'', 1996, Springer-Verlag, pp. 201 and 228–239. define a rational triangle as one with rational sides and rational
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s measured in degrees—in which case the only rational triangle is the rational-sided
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 othe ...
. There are various general properties for an integer triangle, given in the first section below. All other sections refer to classes of integer triangles with specific properties.


General properties for an integer triangle


Integer triangles with given perimeter

Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
: the longest side is shorter than the sum of the other two sides. Each such triple defines an integer triangle that is unique
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
congruence. So the number of integer triangles (up to congruence) with
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 pract ...
''p'' is the number of
partitions Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of ''p'' into three positive parts that satisfy the triangle inequality. This is the integer closest to when ''p'' is
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
and to when ''p'' is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
.Tom Jenkyns and Eric Muller, Triangular Triples from Ceilings to Floors, American Mathematical Monthly 107:7 (August 2000) 634–639 It also means that the number of integer triangles with even numbered perimeters ''p'' = 2''n'' is the same as the number of integer triangles with odd numbered perimeters ''p'' = 2''n'' − 3. Thus there is no integer triangle with perimeter 1, 2 or 4, one with perimeter 3, 5, 6 or 8, and two with perimeter 7 or 10. The
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of the number of integer triangles with perimeter ''p'', starting at ''p'' = 1, is: :0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8 ... This is called
Alcuin's sequence In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: : \frac = x^3 + x^5 + x^6 + 2x^7 + x^8 + 3x^9 + \cdots. The sequence begins with these integers: : 0, 0, 0, 1, 0, ...
.


Integer triangles with given largest side

The number of integer triangles (up to congruence) with given largest side ''c'' and integer triple (''a'', ''b'', ''c'') is the number of integer triples such that ''a'' + ''b'' > ''c'' and ''a'' ≤ ''b'' ≤ ''c''. This is the integer value Ceiling[] * Floor[]. Alternatively, for ''c'' even it is the double triangular number ( + 1) and for ''c'' odd it is the square number, square . It also means that the number of integer triangles with greatest side ''c'' exceeds the number of integer triangles with greatest side ''c'' − 2 by ''c''. The sequence of the number of non-congruent integer triangles with largest side ''c'', starting at ''c'' = 1, is: :1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90 ... The number of integer triangles (up to congruence) with given largest side ''c'' and integer triple (''a'', ''b'', ''c'') that lie on or within a semicircle of diameter ''c'' is the number of integer triples such that ''a'' + ''b'' > ''c'' , ''a2'' + ''b''2 ≤ ''c''2 and ''a'' ≤ ''b'' ≤ ''c''. This is also the number of integer sided obtuse or
right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...
(non-
acute Acute may refer to: Science and technology * Acute angle ** Acute triangle ** Acute, a leaf shape in the glossary of leaf morphology * Acute (medicine), a disease that it is of short duration and of recent onset. ** Acute toxicity, the adverse eff ...
) triangles with largest side ''c''. The sequence starting at ''c'' = 1, is: :0, 0, 1, 1, 3, 4, 5, 7, 10, 13, 15, 17, 22, 25, 30, 33, 38, 42, 48 ... Consequently, the difference between the two above sequences gives the number of acute integer sided triangles (up to congruence) with given largest side ''c''. The sequence starting at ''c'' = 1, is: :1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52 ...


Area of an integer triangle

By
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-century ...
, if ''T'' is the
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 A shape or figure is a graphics, graphical representation of an obje ...
of a triangle whose sides have lengths ''a'', ''b'', and ''c'' then :4T = \sqrt. Since all the terms under the
radical Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
on the right side of the formula are integers it follows that all integer triangles must have an integer value of ''16T2'' and ''T2'' will be rational.


Angles of an integer triangle

By 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 ...
, every angle of an integer triangle has a rational cosine. If the angles of any triangle form an
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
then one of its angles must be 60°. For integer triangles the remaining angles must also have rational cosines and a method of generating such triangles is given below. However, apart from the trivial case of an equilateral triangle, there are no integer triangles whose angles form either a
geometric 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 ca ...
or harmonic progression. This is because such angles have to be rational angles of the form with rational  0 < < 1. But all the angles of integer triangles must have rational cosines and this will occur only when   i.e. the integer triangle is equilateral. The square of each internal
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
of an integer triangle is rational, because the general triangle formula for the internal angle bisector of angle ''A'' is \tfrac where ''s'' is the
semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
(and likewise for the other angles' bisectors).


Side split by an altitude

Any
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
dropped from a vertex onto an opposite side or its extension will split that side or its extension into rational lengths.


Medians

The square of twice any
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
of an integer triangle is an integer, because the general formula for the squared median ''m''a2 to side ''a'' is \tfrac, giving (2''m''a)2 = 2''b''2 + 2''c''2 − ''a''2 (and likewise for the medians to the other sides).


Circumradius and inradius

Because the square of the area of an integer triangle is rational, the square of its
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 ...
is also rational, as is the square of 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. ...
. The ratio of the inradius to the circumradius of an integer triangle is rational, equaling \tfrac for semiperimeter ''s'' and area ''T''. The product of the inradius and the circumradius of an integer triangle is rational, equaling \tfrac. Thus the squared distance between the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
and 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 polyg ...
of an integer triangle, given by
Euler's theorem In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
as ''R''2 − 2''Rr'', is rational.


Heronian triangles

All Heronian triangles can be placed on a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
with each vertex at a lattice point.


General formula

A Heronian triangle, also known as a Heron triangle or a Hero triangle, is a triangle with integer sides and integer area. Every Heronian triangle has sides proportional to :a = n(m^+k^) :b = m(n^+k^) :c = (m+n)(mn-k^) :\text = mn(m+n) :\text = mnk(m+n)(mn-k^) for integers ''m'', ''n'' and ''k'' subject to the constraints: :\gcd = 1 :mn > k^2 \ge m^2n/(2m+n) :m \ge n \ge 1. The proportionality factor is generally a rational \frac where ''q'' = gcd(''a'',''b'',''c'') reduces the generated Heronian triangle to its primitive and p scales up this primitive to the required size.


Pythagorean triangles

A Pythagorean triangle is right-angled and Heronian. Its three integer sides are known as a
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 ...
or Pythagorean triplet or Pythagorean triad.Sierpiński, Wacław. ''
Pythagorean Triangles ''Pythagorean Triangles'' is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw in 1 ...
'', Dover Publications, 2003 (orig. 1962).
All Pythagorean triples (a, b, c) with
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
c which are primitive (the sides having no
common factor 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 ...
) can be generated by : a = m^2 - n^2, \, : b = 2mn, \, : c = m^2 + n^2, \, :\text = m(m+n) \, :\text = mn(m^2-n^2) \, where ''m'' and ''n'' are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
integers and one of them is even with ''m'' > ''n''. Every even number greater than 2 can be the leg of a Pythagorean triangle (not necessarily primitive) because if the leg is given by a=2m and we choose b=(a/2)^2-1=m^2-1 as the other leg then the hypotenuse is c=m^2+1. This is essentially the generation formula above with n set to 1 and allowing m to range from 2 to infinity.


Pythagorean triangles with integer altitude from the hypotenuse

There are no primitive Pythagorean triangles with integer altitude from the hypotenuse. This is because twice the area equals any base times the corresponding height: 2 times the area thus equals both ''ab'' and ''cd'' where ''d'' is the height from the hypotenuse ''c''. The three side lengths of a primitive triangle are coprime, so ''d'' =  is in fully reduced form; since ''c'' cannot equal 1 for any primitive Pythagorean triangle, ''d'' cannot be an integer. However, any Pythagorean triangle with legs ''x'', ''y'' and hypotenuse ''z'' can generate a Pythagorean triangle with an integer altitude, by scaling up the sides by the length of the hypotenuse ''z''. If ''d'' is the altitude, then the generated Pythagorean triangle with integer altitude is given byRichinick, Jennifer, "The upside-down Pythagorean Theorem", ''Mathematical Gazette'' 92, July 2008, 313–317. :(a,b,c,d)=(xz, yz, z^2, xy). \, Consequently, all Pythagorean triangles with legs ''a'' and ''b'', hypotenuse ''c'', and integer altitude ''d'' from the hypotenuse, with gcd(''a, b, c, d'') = 1, which necessarily satisfy both ''a''2 + ''b''2 = c2 and \tfrac+\tfrac=\tfrac, are generated by :a=(m^2-n^2)(m^2+n^2), \, :b=2mn(m^2+n^2), \, :c=(m^2+n^2)^2, \, :d=2mn(m^2-n^2), \, :\text=m(m+n)(m^2+n^2) \, :\text=mn(m^2-n^2)(m^2+n^2)^2 \, for coprime integers ''m'', ''n'' with ''m'' > ''n''.


Heronian triangles with sides in arithmetic progression

A triangle with integer sides and integer area has sides in arithmetic progression
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
the sides are (''b'' – ''d'', ''b'', ''b'' + ''d''), where :b=2(m^2+3n^2)/g, :d=(m^2-3n^2)/g, and where ''g'' is 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 m^2-3n^2, 2mn, and m^2+3n^2.


Heronian triangles with one angle equal to twice another

All Heronian triangles with ''B'' = 2''A'' are generated by either :a = \dfrac, :b = \dfrac, :c = \dfrac, :\text = \dfrac, with integers ''k'', ''s'', ''r'' such that ''s''2 > 3''r''2, or :a = \dfrac \,, :b = q^uv(u^+v^) \,, :c = \dfrac \, , :\text = \dfrac \,, with integers such that and v^2< (7+4\sqrt)u^2. No Heronian triangles with ''B'' = 2''A'' are isosceles or right triangles because all resulting angle combinations generate angles with non-rational
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
s, giving a non-rational area or side.


Isosceles Heronian triangles

All
isosceles 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 ...
Heronian triangles are decomposable. They are formed by joining two congruent Pythagorean triangles along either of their common legs such that the equal sides of the isosceles triangle are the hypotenuses of the Pythagorean triangles, and the base of the isosceles triangle is twice the other Pythagorean leg. Consequently, every Pythagorean triangle is the building block for two isosceles Heronian triangles since the join can be along either leg. All pairs of isosceles Heronian triangles are given by rational multiples ofSastry, K. R. S.
"Construction of Brahmagupta n-gons"
''Forum Geometricorum'' 5 (2005): 119–126.
:a=2(u^2-v^2), :b=u^2+v^2, :c=u^2+v^2, and :a=4uv, :b=u^2+v^2, :c=u^2+v^2, for coprime integers ''u'' and ''v'' with ''u'' > ''v'' and ''u'' + ''v'' odd.


Heronian triangles whose perimeter is four times a prime

It has been shown that a Heronian triangle whose perimeter is four times a
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
is uniquely associated with the prime and that the prime is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
to 1 or 3 modulo 8. It is well known that such a prime p can be uniquely partitioned into integers m and n such that p=m^2+2n^2 (see Euler's idoneal numbers). Furthermore, it has been shown that such Heronian triangles are primitive since the smallest side of the triangle has to be equal to the prime that is one quarter of its perimeter. Consequently, all primitive Heronian triangles whose perimeter is four times a prime can be generated by :a=m^2+2n^2 :b=m^2+4n^2 :c=2(m^2+n^2) :\text=2a=2(m^2+2n^2) :\text=2mn(m^2+2n^2) for integers m and n such that m^2+2n^2 is a prime. Furthermore, the factorization of the area is 2mnp where p=m^2+2n^2 is prime. However the area of a Heronian triangle is always divisible by 6. This gives the result that apart from when m=1 and n=1, which gives p=3, all other parings of m and n must have m odd with only one of them divisible by 3.


Heronian triangles with rational angle bisectors

If in a Heronian triangle the angle bisector w_a of the angle \alpha, the angle bisector w_b of the angle \beta and the angle bisector w_c of the angle \gamma have a rational relationship with the three sides then not only \alpha, \beta, \gamma but also \frac, \frac and \frac must be Heronian angles. Namely, if both angles \frac and \frac are Heronian then \frac, the complement of \frac + \frac, must also be a Heronian angle, so that all three angle-bisectors are rational. This is also evident if one multiplies: :w_a = \frac \quad w_b = \frac \quad w_c = \frac together. Namely, through this one obtains: :w_a \cdot w_b \cdot w_c = \frac, where s denotes the semi-perimeter, and J the area of the triangle. All Heronian triangles with rational angle bisectors are generated by :a = mn(p^2+q^2) :b = pq(m^2+n^2) :c = (mq+np)(mp-nq) :\text=s=(a+b+c)/2=mp(mq+np) :s-a=mq(mp-nq) :s-b=np(mp-nq) :s-c=nq(mq+np) :\text=J=mnpq(mq+np)(mp-nq) where m, n, p, q are such that :m = t^2-u^2 :n=2tu :p = v^2-w^2 :q=2vw where t, u, v, w are arbitrary integers such that :t and u coprime, :v and w coprime.


Heronian triangles with integer inradius and exradii

There are infinitely many decomposable, and infinitely many indecomposable, primitive Heronian (non-Pythagorean) triangles with integer radii for the
incircle 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. ...
and each
excircle 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. ...
.Li Zhou, "Primitive Heronian Triangles With Integer Inradius and Exradii", ''Forum Geometricorum'' 18, 2018, pp. 71–77. A family of decomposible ones is given by :a=4n^2 :b=(2n+1)(2n^2 -2n+1) :c=(2n-1)(2n^2 +2n+1) :r=2n-1 :r_a=2n+1 :r_b=2n^2 :r_c=\text=2n^2(2n-1)(2n+1); and a family of indecomposable ones is given by :a=5(5n^2 +n-1) :b=(5n+3)(5n^2-4n+1) :c=(5n-2)(5n^2 +6n+2) :r=5n-2 :r_a=5n+3 :r_b=5n^2+n-1 :r_c=\text=(5n-2)(5n+3)(5n^2 +n-1).


Heronian triangles as faces of a tetrahedron

There exist
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
having integer-valued
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
and Heron triangles as
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
. One example has one edge of 896, the opposite edge of 190, and the other four edges of 1073; two faces have areas of 436800 and the other two have areas of 47120, while the volume is 62092800.


Heronian triangles in a 2D lattice

A 2D
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at (''x, y'') where ''x'' and ''y'' range over all positive and negative integers. A lattice triangle is any triangle drawn within a 2D lattice such that all vertices lie on lattice points. By
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...
a lattice triangle has a rational area that either is an integer or a
half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include numbers ...
(has a denominator of 2). If the lattice triangle has integer sides then it is Heronian with integer area. Furthermore, it has been proved that all Heronian triangles can be drawn as lattice triangles. Consequently, an integer triangle is Heronian if and only if it can be drawn as a lattice triangle. There are infinitely many primitive Heronian (non-Pythagorean) triangles which can be placed on an integer lattice with all vertices, the
incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
, and all three
excenter 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. ...
s at lattice points. Two families of such triangles are the ones with parametrizations given above at #Heronian triangles with integer inradius and exradii.


Integer automedian triangles

An automedian triangle is one whose medians are in the same proportions (in the opposite order) as the sides. If ''x'', ''y'', and ''z'' are the three sides of a right triangle, sorted in increasing order by size, and if 2''x'' < ''z'', then ''z'', ''x'' + ''y'', and ''y'' − ''x'' are the three sides of an automedian triangle. For instance, the right triangle with side lengths 5, 12, and 13 can be used in this way to form the smallest non-trivial (i.e., non-equilateral) integer automedian triangle, with side lengths 13, 17, and 7.. Consequently, using
Euclid's formula 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 ...
, which generates primitive Pythagorean triangles, it is possible to generate primitive integer automedian triangles as :a=, m^2-2mn-n^2, :b=m^2+2mn-n^2 :c=m^2+n^2 with m and n coprime and m+n odd, and n  (if the quantity inside the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
signs is negative) or  m>(2+\sqrt)n (if that quantity is positive) to satisfy the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
. An important characteristic of the automedian triangle is that the squares of its sides form an
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
. Specifically, c^2-a^2 = b^2-c^2 so 2c^2 = a^2+b^2.


Integer triangles with specific angle properties


Integer triangles with a rational angle bisector

A triangle family with integer sides a,b,c and with rational bisector d of angle ''A'' is given by :a = 2(k^2-m^2), :b = (k-m)^2, :c = (k+m)^2, :d = \tfrac, with integers k>m>0.


Integer triangles with integer ''n''-sectors of all angles

There exist infinitely many non- similar triangles in which the three sides and the bisectors of each of the three angles are integers. There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers. However, for ''n'' > 3 there exist no triangles in which the three sides and the (''n'' – 1) ''n''-sectors of each of the three angles are integers.De Bruyn,Bart, "On a Problem Regarding the n-Sectors of a Triangle", ''Forum Geometricorum'' 5, 2005: pp. 47–52.
/ref>


Integer triangles with one angle with a given rational cosine

Some integer triangles with one angle at vertex ''A'' having given rational cosine ''h'' / ''k'' (''h'' < 0 or > 0; ''k'' > 0) are given by :a = p^2-2pqh+q^2k^2, :b = p^2-q^2k^2, :c = 2qk(p-qh), where ''p'' and ''q'' are any coprime positive integers such that ''p'' > ''qk''.


Integer triangles with a 60° angle (angles in arithmetic progression)

All integer triangles with a 60° angle have their angles in an arithmetic progression. All such triangles are proportional to: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> :a = 4mn, :b = 3m^2+n^2, :c = 2mn+, 3m^2-n^2, with coprime integers ''m'', ''n'' and 1 ≤ ''n'' ≤ ''m'' or 3''m'' ≤ ''n''. From here, all primitive solutions can be obtained by dividing ''a'', ''b'', and ''c'' by their greatest common divisor. Integer triangles with a 60° angle can also be generated by :a = m^2-mn+n^2, :b = 2mn-n^2, :c = m^2-n^2, with coprime integers ''m'', ''n'' with 0 < ''n'' < ''m'' (the angle of 60° is opposite to the side of length ''a''). From here, all primitive solutions can be obtained by dividing ''a'', ''b'', and ''c'' by their greatest common divisor (e.g. an equilateral triangle solution is obtained by taking and , but this produces ''a'' = ''b'' = ''c'' = 3, which is not a primitive solution). See also Burn, Bob, "Triangles with a 60° angle and sides of integer length", ''Mathematical Gazette'' 87, March 2003, 148–153.Read, Emrys, "On integer-sided triangles containing angles of 120° or 60°", ''Mathematical Gazette'' 90, July 2006, 299−305. More precisely, If m \equiv -n \ (\text \ 3), then gcd(a,b,c) = 3, otherwise gcd(a,b,c) = 1. Two different pairs (m, n) and (m, m-n) generate the same triple. Unfortunately the two pairs can both be of gcd = 3, so we can't avoid duplicates by simply skipping that case. Instead, duplicates can be avoided by n going only till m/2. We still need to divide by 3 if gcd = 3. The only solution for n = m/2 under the above constraints is (3,3,3) \equiv (1,1,1) for m=2, n=1. With this additional n \leq m/2 constraint all triples can be generated uniquely. An
Eisenstein triple Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 or 120 degrees. The relation of such triangles to the Eisens ...
is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 degrees.


Integer triangles with a 120° angle

Integer triangles with a 120° angle can be generated by :a = m^2 + mn + n^2, :b = 2mn+n^2, :c = m^2 - n^2, with coprime integers ''m'', ''n'' with 0 < ''n'' < ''m'' (the angle of 120° is opposite to the side of length ''a''). From here, all primitive solutions can be obtained by dividing ''a'', ''b'', and ''c'' by their greatest common divisor. The smallest solution, for ''m'' = 2 and ''n'' = 1, is the triangle with sides (3,5,7). See also. More precisely, If m \equiv n \ (\text \ 3), then \gcd(a,b,c) = 3, otherwise \gcd(a,b,c) = 1. Since the biggest side ''a'' can only be generated with a single (m, n) pair, each primitive triple can be generated in precisely two ways: once directly with gcd = 1, and once indirectly with gcd = 3. Therefore, in order to generate all primitive triples uniquely, one can just add additional m \not\equiv n \ (\text \ 3) condition.


Integer triangles with one angle equal to an arbitrary rational number times another angle

For positive coprime integers ''h'' and ''k'', the triangle with the following sides has angles h \alpha, k \alpha, and \pi - (h+k) \alpha and hence two angles in the ratio ''h'' : ''k'', and its sides are integers: :a = q^ \frac = q^k \cdot\sum_(-1)^\binomp^(q^2-p^2)^i, :b = q^ \frac = q^h \cdot\sum_(-1)^\binomp^(q^2-p^2)^i, :c = q^ \frac = \sum_(-1)^\binomp^(q^2-p^2)^i, where \alpha = \cos^ \!\frac and ''p'' and ''q'' are any coprime integers such that \cos \frac < \frac < 1.


Integer triangles with one angle equal to twice another

With angle ''A'' opposite side a and angle ''B'' opposite side b, some triangles with ''B'' = 2''A'' are generated byDeshpande,M. N., "Some new triples of integers and associated triangles", ''Mathematical Gazette'' 86, November 2002, 464–466. :a = n^2, :b = mn, :c = m^2 - n^2, with integers ''m'', ''n'' such that 0 < ''n'' < ''m'' < 2''n''. All triangles with ''B'' = 2''A'' (whether integer or not) satisfy a(a+c) = b^2.


Integer triangles with one angle equal to 3/2 times another

The equivalence class of similar triangles with B = \tfracA are generated by :a=mn^3, :b=n^2(m^2-n^2), :c=(m^2 - n^2)^2 - m^2 n^2, with integers m, n such that 0<\varphi n, where \varphi is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
\varphi = \frac\approx 1.61803. All triangles with B = \tfracA (whether with integer sides or not) satisfy (b^-a^)(b^-a^+bc) = a^c^.


Integer triangles with one angle three times another

We can generate the full equivalence class of similar triangles that satisfy ''B'' = 3''A'' by using the formulas :a=n^, \, :b=n(m^-n^), \, :c=m(m^-2n^), \, where m and n are integers such that \sqrtn < m < 2n. All triangles with ''B'' = 3''A'' (whether with integer sides or not) satisfy ac^2 = (b-a)^(b+a).


Integer triangles with three rational angles

The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the
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 othe ...
. This is because integer sides imply three rational cosines by 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 ...
, and by
Niven's theorem In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of ''θ'' in the interval 0° ≤ ''θ'' ≤ 90° for which the sine of ''θ'' degrees is also a rational number ...
a rational cosine coincides with a rational angle if and only if the cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles.


Integer triangles with integer ratio of circumradius to inradius

Conditions are known in terms of
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s for an integer triangle to have an integer ratio ''N'' of the
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 ...
to 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. ...
. The smallest case, that of the
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 othe ...
, has ''N'' = 2. In every known case, ''N'' ≡ 2 (mod 8)—that is, ''N'' – 2 is divisible by 8.


5-Con triangle pairs

A 5-Con triangle pair is a pair of triangles that are similar but not
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
and that share three angles and two sidelengths. Primitive integer 5-Con triangles, in which the four distinct integer sides (two sides each appearing in both triangles, and one other side in each triangle) share no prime factor, have triples of sides :(x^3, x^2y, xy^2) and (x^2y, xy^2, y^3) for positive coprime integers ''x'' and ''y''. The smallest example is the pair (8, 12, 18), (12, 18, 27), generated by ''x'' = 2, ''y'' = 3.


Particular integer triangles

*The only triangle with consecutive integers for sides and area has sides (3, 4, 5) and area 6. *The only triangle with consecutive integers for an altitude and the sides has sides (13, 14, 15) and altitude from side 14 equal to 12. *The (2, 3, 4) triangle and its multiples are the only triangles with integer sides in arithmetic progression and having the complementary exterior angle property.Mitchell, Douglas W., "The 2:3:4, 3:4:5, 4:5:6, and 3:5:7 triangles", ''Mathematical Gazette'' 92, July 2008. This property states that if angle C is obtuse and if a segment is dropped from B meeting perpendicularly AC extended at P, then ∠CAB=2∠CBP. *The (3, 4, 5) triangle and its multiples are the only integer right triangles having sides in arithmetic progression. *The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression. *The (3, 5, 7) triangle and its multiples are the only triangles with a 120° angle and having integer sides in arithmetic progression. *The only integer triangle with area = semiperimeterMacHale, D., "That 3,4,5 triangle again", ''Mathematical Gazette'' 73, March 1989, 14−16. has sides (3, 4, 5). *The only integer triangles with area = perimeter have sides (5, 12, 13), (6, 8, 10), (6, 25, 29), (7, 15, 20), and (9, 10, 17). Of these the first two, but not the last three, are right triangles. *There exist integer triangles with three rational
medians The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
. The smallest has sides (68, 85, 87). Others include (127, 131, 158), (113, 243, 290), (145, 207, 328) and (327, 386, 409). *There are no isosceles Pythagorean triangles. *The only primitive Pythagorean triangles for which the square of the perimeter equals an integer multiple of the area are (3, 4, 5) with perimeter 12 and area 6 and with the ratio of perimeter squared to area being 24; (5, 12, 13) with perimeter 30 and area 30 and with the ratio of perimeter squared to area being 30; and (9, 40, 41) with perimeter 90 and area 180 and with the ratio of perimeter squared to area being 45.Goehl, John F. Jr., "Pythagorean triangles with square of perimeter equal to an integer multiple of area", ''Forum Geometricorum'' 9 (2009): 281–282.
/ref> *There exists a unique (up to similitude) pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. The unique pair consists of the (377, 135, 352) triangle and the (366, 366, 132) triangle. There is no pair of such triangles if the triangles are also required to be primitive integral triangles. The authors stress the striking fact that the second assertion can be proved by an elementary argumentation (they do so in their appendix A), whilst the first assertion needs modern highly non-trivial mathematics.


See also

*
Robbins pentagon In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers. History Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a ...
, a cyclic pentagon with integer sides and integer area *
Euler brick In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick i ...
, a cuboid with integer edges and integer face diagonals * Tetrahedron#Integer tetrahedra


References

{{DEFAULTSORT:Integer Triangle Arithmetic problems of plane geometry Discrete geometry Squares in number theory Types of triangles