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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. Integer triangles with given largest side 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. 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. This is the integer closest to p 2⁄ 48 when p is even and to ( p + 3) 2⁄ 48 when p is odd. 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. Each such triple defines an integer triangle that is unique up to congruence. General properties for an integer triangle Integer triangles with given perimeter Īny 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.

5 Integer triangles with integer ratio of circumradius to inradius.

4.5 Integer triangles with three rational angles.

4.4.3 Integer triangles with one angle three times another.

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

4.4.1 Integer triangles with one angle equal to twice another.

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

4.3.2 Integer triangles with a 120° angle.

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

4.3 Integer triangles with one angle with a given rational cosine.

4.2 Integer triangles with integer n-sectors of all angles.

4.1 Integer triangles with a rational angle bisector.

4 Integer triangles with specific angle properties.

2.10 Heronian triangles in a 2D lattice.

2.9 Heronian triangles as faces of a tetrahedron.

2.8 Heronian triangles with integer inradius and exradii.

2.7 Heronian triangles with rational angle bisectors.

2.6 Heronian triangles whose perimeter is four times a prime.

2.4 Heronian triangles with one angle equal to twice another.

2.3 Heronian triangles with sides in arithmetic progression.

2.2.1 Pythagorean triangles with integer altitude from the hypotenuse.

1.2 Integer triangles with given largest side.

1.1 Integer triangles with given perimeter.

1 General properties for an integer triangle.

All other sections refer to classes of integer triangles with specific properties.

There are various general properties for an integer triangle, given in the first section below. 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. 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. A Heronian triangle with sidelengths c, e and b + d, and height a, all integers.Īn integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers.