An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles.
Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a Heronian triangle (a triangle with integral or rational side lengths and area); Conway and Guy (1996) define a rational triangle as one with rational sides and rational angles measured in degrees—the only such triangles are rational-sided equilateral triangles.Conway, J. H., and Guy, R. K., "The only rational triangle", in The Book of Numbers, 1996, Springer-Verlag, pp. 201 and 228–239.
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.Tom Jenkyns and Eric Muller, Triangular Triples from Ceilings to Floors, American Mathematical Monthly 107:7 (August 2000) 634–639Ross Honsberger, Mathematical Gems III, pp. 39–37 It also means that the number of integer triangles with even numbered perimeters is the same as the number of integer triangles with odd numbered perimeters 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 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.
Integer triangles with given largest side The number of integer triangles (up to congruence) with given largest side c and integer triple is the number of integer triples such that and This is the integer value Alternatively, for c even it is the double triangular number 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: 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 + b2 ≤ c2 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: 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, 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 16T2 an integer and T2 will be rational.
Angles of an integer triangle
By the law of cosines, every angle of an integer triangle has a rational cosine. Every angle of an integer right triangle also has rational sine (see Pythagorean triple).
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 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 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 ma2 to side a is , giving (2ma)2 = 2b2 + 2c2 − a2 (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 for semiperimeter s and area T.
The product of the inradius and the circumradius of an integer triangle is rational, equaling
Thus the squared distance between the incenter and the circumcenter of an integer triangle, given by Euler's theorem as is rational.
Heronian triangles
A Heronian triangle, also known as a Heron triangle or a Hero triangle, is a triangle with integer sides and integer area.
All Heronian triangles can be placed on a lattice with each vertex at a lattice point.Yiu, P., "Heronian triangles are lattice triangles", American Mathematical Monthly 108 (2001), 261–263. Furthermore, if an integer triangle can be place on a lattice with each vertex at a lattice point it must be Heronian.
General formula
Every Heronian triangle has sides proportional toCarmichael, R. D. The Theory of Numbers and Diophantine Analysis. New York: Dover, 1952.
for integers m, n and k subject to the constraints:
The proportionality factor is generally a rational where q = gcd(a,b,c) reduces the generated Heronian triangle to its primitive and 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.Sierpiński, Wacław. Pythagorean Triangles, Dover Publications, 2003 (orig. 1962). All Pythagorean triples with hypotenuse 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 and we choose as the other leg then the hypotenuse is . This is essentially the generation formula above with set to 1 and allowing 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 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.
Consequently, all Pythagorean triangles with legs a and b, hypotenuse c, and integer altitude d from the hypotenuse, with , which necessarily satisfy both a2 + b2 = c2 and , are generated byVoles, Roger, "Integer solutions of a−2+b−2=d−2", Mathematical Gazette 83, July 1999, 269–271.
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 and
Heronian triangles with one angle equal to twice another
All Heronian triangles with B = 2A are generated byMitchell, Douglas W., "Heron triangles with ∠B=2∠A", Mathematical Gazette 91, July 2007, 326–328. either
with integers k, s, r such that or
with integers such that and
No Heronian triangles with B = 2A 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 the following side lengths:Sastry, K. R. S., "Construction of Brahmagupta n-gons" , Forum Geometricorum 5 (2005): 119–126.
for coprime integers of opposite parity and , with .
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 or modulo .Yiu, P., "CRUX, Problem 2331, Proposed by Paul Yiu" , Memorial University of Newfoundland (1998): 175-177Yui, P. and Taylor, J. S., "CRUX, Problem 2331, Solution" Memorial University of Newfoundland (1999): 185-186 It is well known that such a prime can be uniquely partitioned into integers and such that (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 and such that is a prime.
Furthermore, the factorization of the area is where is prime. However the area of a Heronian triangle is always divisible by . This gives the result that apart from when and which gives all other parings of and must have odd with only one of them divisible by .
Heronian triangles with rational angle bisectors
If in a Heronian triangle the angle bisector of the angle , the angle bisector of the angle and the angle bisector of the angle have a rational relationship with the three sides then not only but also , and must be Heronian angles. Namely, if both angles and are Heronian then , the complement of , must also be a Heronian angle, so that all three angle-bisectors are rational. This is also evident if one multiplies:
together. Namely, through this one obtains:
where denotes the semi-perimeter, and the area of the triangle.
All similarity classes of Heronian triangles with rational angle bisectors are generated byHermann Schubert, "Die Ganzzahligkeit in der Algebraischen Geometrie", Leipzig, 1905
where are such that
where are arbitrary integers such that and coprime, and coprime.