Integer triangle

Every angle of an integer right triangle also has rational sine (see Pythagorean triple).

But all the angles of integer triangles must have rational cosines and this will occur only when

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

where q = gcd(a,b,c) reduces the generated Heronian triangle to its primitive and

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

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.

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.

All pairs of isosceles Heronian triangles are given by rational multiples of[15] and for coprime integers u and v with u > v and u + v odd.

Consequently, all primitive Heronian triangles whose perimeter is four times a prime can be generated by for integers

All Heronian triangles with rational angle bisectors are generated by[18] where

are arbitrary integers such that There are infinitely many decomposable, and infinitely many indecomposable, primitive Heronian (non-Pythagorean) triangles with integer radii for the incircle and each excircle.[19]: Thms.

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).

5 An automedian triangle is one whose medians are in the same proportions (in the opposite order) as the sides.

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.

An important characteristic of the automedian triangle is that the squares of its sides form an arithmetic progression.

There exist infinitely many non-similar triangles in which the three sides and the bisectors of each of the three angles are integers.

[25] There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers.

[25] 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.

All primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor.

All such triangles are proportional to:[5] with coprime integers m, n and 1 ≤ n ≤ m or 3m ≤ n. From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor.

Unfortunately the two pairs can both have a gcd of 3, so we can't avoid duplicates by simply skipping that case.

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.

From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor.

Therefore, in order to generate all primitive triples uniquely, one can just add additional

[citation needed] For positive coprime integers h and k, the triangle with the following sides has angles

We can generate the full equivalence class of similar triangles that satisfy B = 3A by using the formulas[34] where