Integer triangle

An integer triangle or integral triangle is a triangle all of whose sides have lengths which are integers. A rational triangle has all sides with rational length; any 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 rational and integer triangles.

Pythagorean triangles

Main articles: Pythagorean triangle, Pythagorean triple

A Pythagorean triangle has a right angle and three integer sides, which are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad . All Pythagorean triples $(a,b,c)$ with hypotenuse $c$ which are primitive (the sides having no common factor) can be generated by

a=$m^{2}-n^{2}$,

$b=2mn$,

$c=m^{2}+n^{2}$,

where $m$ and $n$ are coprime integers with $m>n$.

Pythagorean triangles with integer altitude from the hypotenuse

All primitive Pythagorean triangles with legs $a,b$, hypotenuse $c$, and integer altitude $d$ from the hypotenuse, which necessarily have both $a^{2}+b^{2}=c^{2}$ and ${\tfrac {1}{a^{2}}}+{\tfrac {1}{b^{2}}}={\tfrac {1}{d^{2}}}$, are generated by

$a=(v^{2}-u^{2})(v^{2}+u^{2})$,

$b=2uv(v^{2}+u^{2})$,

$c=(v^{2}+u^{2})^{2}$,

$d=2uv(v^{2}-u^{2})$,

for coprime integers $u,v$ with $v>u$.

Furthermore, any Pythagorean triangle with legs $x,y$ and hypotenuse $z$ can generate another Pythagorean triangle, this one with integer altitude $d$ from the hypotenuse $c$, by

$(a,b,d)=(xz,yz,xy)$.

Heronian triangles

Main article: Heronian triangle

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 generated as multiples of

$a=n(m^{2}+k^{2})$,

$b=m(n^{2}+k^{2})$,

$c=(m+n)(mn-k^{2})$,

for rational numbers $m,n$.

Integer triangles with a 60° angle

All primitive integer triangles with a 60° angle can be generated by

$a=m^{2}+mn+n^{2}$

$b=m^{2}+2mn$

$c=n^{2}+2mn$

or

$c=m^{2}-n^{2}$

with coprime integers $m,n,$ $0<n<m$, $m$ not congruent to $n$ (mod 3).

Integer triangles with a 120° angle

All primitive 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 $m,n,$ $0<n<m$, $m$ not congruent to $n$ (mod 3).

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=2A are generated by

$a=n^{2}$,

$b=mn$,

$c=m^{2}-n^{2}$,

with integers $m,n$ such that $0<n<m<2n$.

All triangles with B=2A have $a(a+c)=b^{2}$.

Heronian triangles with one angle equal to twice another

All Heronian triangles with B=2A are generated by either

$a={\tfrac {k^{2}(s^{2}+r^{2})^{2}}{4}}$,

$b={\tfrac {k^{2}(s^{4}-r^{4})}{2}}$,

$c={\tfrac {k^{2}(3s^{4}-10s^{2}r^{2}+3r^{4})}{4}}$,

$Area={\tfrac {k^{2}csr(s^{2}-r^{2})}{2}}$,

with integers $k,s,r$ such that $s^{2}>3r^{2}$, or

$a={\tfrac {q^{2}(u^{2}+v^{2})^{2}}{4}}$,

$b=q^{2}uv(u^{2}+v^{2})$,

$c={\tfrac {q^{2}(14u^{2}v^{2}-u^{4}-v^{4})}{4}}$,

$Area={\tfrac {q^{2}cuv(v^{2}-u^{2})}{2}}$,

with integers $q,u,v$ such that $v>u$ and $v^{2}<(7+4{\sqrt {3}})u^{2}$.

No triangles with B=2A are Pythagorean.

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

Some triangles with B=${\tfrac {3}{2}}$A 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 $0<\varphi n<m<2n$, where $\varphi$ is the golden ratio $\varphi ={\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$.

Integer triangles with one angle three times another

Some triangles with B=3A are generated by

$a=8m^{3}n^{3}$,

$b=2mn(m^{2}-n^{2})^{2}$,

$c=(m^{4}+n^{4}-6m^{2}n^{2})(m^{2}+n^{2})$,

with integers $m,n$ such that $0<(1+{\sqrt {2}})n<m<(2+{\sqrt {3}})n$.

In addition, some triangles with B=3A are generated by

$a=n^{3}$,

$b=n(m^{2}-n^{2})$,

$c=m(m^{2}-2n^{2})$,

with integers $m,n$ such that $2n^{2}<m^{2}<4n^{2}$.

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

$a=2(k^{2}-m^{2})$,

$b=(k-m)^{2}$,

$c=(k+m)^{2}$,

$d={\tfrac {2km(k^{2}-m^{2})}{k^{2}+m^{2}}}$,

with integers $k>m>0$.

Miscellaneous facts about 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 complimentary exterior angle property.

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 60° angle and having integer sides in arithmetic progression.

The only integer triangle with area=semiperimeter 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 are no equilateral Heronian triangles.