Integer triangle
An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rational triangle can be integrally rescaled (can have all sides multiplied by the same integer, namely a common multiple 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; Somos uses it to refer to triangles whose ratios of sides are rational; Conway and Guy define a rational triangle as one with rational sides and rational angles measured in degrees—in which case the only rational triangle is the rational-sided equilateral triangle. 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: 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 congruence. So the number of integer triangles (up to congruence) with perimeter "p" is the number of partitions of "p" into three positive parts that satisfy the triangle inequality. This is the integer closest to when "p" is even and to when "p" is odd. 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 of the number of integer triangles with perimeter "p", starting at "p" = 1, is: This is called Alcuin's sequence. 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 . 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: 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" , "a" + "b" ≤ "c" and "a" ≤ "b" ≤ "c". This is also the number of integer sided obtuse or right (non-acute) triangles with largest side "c". The sequence starting at "c" = 1, is: 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: Area of an integer triangle. By Heron 's formula, if "T" is the area of a triangle whose sides have lengths "a", "b", and "c" then Since all the terms under the radical on the right side of the formula are integers it follows that all integer triangles must have an integer value of "16T" and "T" will be rational. Angles of an integer triangle. By the law of cosines, every angle of an integer triangle has a rational cosine. If the angles of any triangle form an arithmetic progression 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 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 of an integer triangle is rational, because the general triangle formula for the  internal angle bisector of angle "A" is formula_2 where "s" is the semiperimeter (and likewise for the other angles' bisectors). Side split by an altitude. Any altitude 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 of an integer triangle is an integer, because the general formula for the squared median "m" to side "a" is formula_3, giving (2"m") = 2"b" + 2"c" − "a" (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 is also rational, as is the square of the inradius. The ratio of the inradius to the circumradius of an integer triangle is rational, equaling formula_4 for semiperimeter "s" and area "T". The product of the inradius and the circumradius of an integer triangle is rational, equaling formula_5 Thus the squared distance between the incenter and the circumcenter of an integer triangle, given by Euler's theorem as "R" − 2"Rr", is rational. Heronian triangles. All Heronian triangles can be placed on a lattice 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 for integers "m", "n" and "k" subject to the constraints: The proportionality factor is generally a rational formula_14 where "q" = gcd("a","b","c") reduces the generated Heronian triangle to its primitive and formula_15 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 or Pythagorean triplet or Pythagorean  triad. All Pythagorean triples formula_16 with hypotenuse formula_17 which are primitive (the sides having no common factor) can be generated by where "m" and "n" are coprime 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 formula_23 and we choose formula_24 as the other leg then the hypotenuse is formula_25. This is essentially the generation formula above with formula_26 set to 1 and allowing formula_27 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 by 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" + "b" = c and formula_29, are generated by 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 the sides are ("b" – "d", "b", "b" + "d"), where and where "g" is the greatest common divisor of formula_38 formula_39 and formula_40 Heronian triangles with one angle equal to twice another. All Heronian triangles with "B" = 2"A" are generated by either with integers "k", "s", "r" such that "s" > 3"r", or with integers such that and formula_49 No Heronian triangles with "B" = 2"A" are isosceles or right triangles because all resulting angle combinations generate angles with non-rational sines, giving a non-rational area or side. Isosceles Heronian triangles. All isosceles 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 of and 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 is uniquely associated with the prime and that the prime is congruent to formula_56 or formula_57 modulo formula_58. It is well known that such a prime formula_15 can be uniquely partitioned into integers formula_27 and formula_26 such that formula_62 (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 for integers formula_27 and formula_26 such that formula_70 is a prime. Furthermore, the factorization of the area is formula_71 where formula_62 is prime. However the area of a Heronian triangle is always divisible by formula_73. This gives the result that apart from when formula_74 and formula_75 which gives formula_76 all other parings of formula_27 and formula_26 must have formula_27 odd with only one of them divisible by formula_57. 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 and each excircle. A family of decomposible ones is given by and a family of indecomposable ones is given by Heronian triangles as faces of a tetrahedron. There exist tetrahedra having integer-valued volume and Heron triangles as faces. 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 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 a lattice triangle has a rational area that either is an integer or a half-integer (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, and all three excenters 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 2x" < "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, which generates primitive Pythagorean triangles, it is possible to generate primitive integer automedian triangles as with formula_27 and formula_26 coprime and formula_90  odd, and formula_91  (if the quantity inside the absolute value signs is negative) or  formula_92 (if that quantity is positive) to satisfy the triangle inequality. An important characteristic of the automedian triangle is that the squares of its sides form an arithmetic progression. Specifically, formula_93 so formula_94 Integer triangles with specific angle properties. Integer triangles with a rational angle bisector. A triangle family with integer sides formula_95 and with rational bisector formula_96 of angle "A" is given by with integers formula_101. 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. 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  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: 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 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 More precisely, If formula_111, then formula_112, otherwise formula_113. Two different pairs formula_114 and formula_115 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 formula_26 going only till formula_117. We still need to divide by 3 if gcd = 3. The only solution for formula_118 under the above constraints is formula_119 for formula_120. With this additional formula_121 constraint all triples can be generated uniquely. An Eisenstein triple 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 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 formula_125, then formula_126, otherwise formula_127. Since the biggest side "a" can only be generated with a single formula_114 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 formula_129 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 formula_130, formula_131, and formula_132 and hence two angles in the ratio "h" : "k", and its sides are integers: where formula_136 and "p" and "q" are any coprime integers such that formula_137. Integer triangles with one angle equal to twice another. With angle "A" opposite side formula_138 and angle "B" opposite side formula_139, some triangles with  B = 2"A" are generated by with integers "m", "n" such that 0 < "n" < "m" < 2"n". All triangles with "B" = 2"A" (whether integer or not) satisfy formula_143 Integer triangles with one angle equal to 3/2 times another. The equivalence class of similar triangles with formula_144 are generated by with integers formula_148 such that formula_149, where formula_150 is the golden ratio formula_151. All triangles with formula_144 (whether with integer sides or not) satisfy formula_153 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 where formula_27 and formula_26 are integers such that formula_159. All triangles with B" = 3"A" (whether with integer sides or not) satisfy formula_160 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. This is because integer sides imply three rational cosines by the law of cosines, and by Niven's theorem 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 curves for an integer triangle to have an integer ratio "N" of the circumradius to the inradius. The smallest case, that  of the equilateral triangle, 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 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 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.