## Integer triangles with 120° angle

Law of cosines
https://en.wikipedia.org/wiki/Law_of_cosines

$c^2 \; = \; a^2 \; + \; b^2 \; - \; 2 \, a \, b \, \cos \, \gamma$

If   $\gamma \; = \; 120^\circ$   —>   $\cos 120^\circ \; = \; -0.5$

then,     $c^2 \; = \; a^2 \; + \; b^2 \; + \; a \, b$

To find triples   $(a, b, c)$   that satisfy the relation   $c^2 \; = \; a^2 \; + \; b^2 \; + \; a \, b$

We use the parametric equations :

$a \; = \; m^2 \; - \; n^2$
$b \; = \; 2 \, m \, n \; + \; n^2$
$c \; = \; m^2 \; + \; m \, n \; + \; n^2$

where   $m$   and   $n$   are integers and   $m \; > \; n$

$(m^2 - n^2 )^2 \; + \; (m^2 - n^2 ) \,(2 \, m \, n + n^2 ) \; + \; (2 \, m \, n + n^2 )^2$
$= \; (m^2 + m \, n + n^2 )^2$

Perimeter   $P$ :

$P \; = \; m^2 \; - \; n^2 \; + \; 2 \, m \, n \; + \; n^2 \; + \; m^2 \; + \; m \, n \; + \; n^2$
$P \; = \; 2 \, m^2 \; + \; 3 \, m \, n \; + \; n^2$
$P \; = \; (2 \, m \; + \; n) \, (m \; + \; n)$

The first few primitive triples are:

$(3, \; 5, \; 7)$
$(5, \; 16, \; 19)$
$(7, \; 8, \; 13)$
$(7, \; 33, \; 37)$
$(9, \; 56, \; 61)$
$(11, \; 24, \; 31)$
$(11, \; 85, \; 91)$
$(13, \; 35, \; 43)$
$(17, \; 63, \; 73)$
$(19, \; 80, \; 91)$

In the case where   $(a, \; b)$   are consecutive integers:

The first few triples such that   $(a, \; a+1, \; c)$

$(7, \; 8, \; 13)$
$(104, \; 105, \; 181)$
$(1455, \; 1456, \; 2521)$
$(20272, 20273, \; 35113)$
$(282359, \; 282360, \; 489061)$
$(3932760, \; 3932761, \; 6811741)$
$(54776287, \; 54776288, \; 94875313)$
$(762935264, \; 762935265, \; 1321442641)$
$(10626317415, \; 10626317416, \; 18405321661)$
$(148005508552, \; 148005508553, \; 256353060613)$
$(2061450802319, \; 2061450802320, \; 3570537526921)$
$(28712305723920, \; 28712305723921, \; 49731172316281)$
$(399910829332567, \; 399910829332568, \; 692665874901013)$
$(5570039304932024, \; 5570039304932025, \; 9647591076297901)$

The first few triples such that   $(a, \; a+2, \; c)$

$(3, \; 5, \; 7)$
$(14, \; 16, \; 26)$
$(55, \; 57, \; 97)$
$(208, \; 210, \; 362)$
$(779, \; 781, \; 1351)$
$(2910, \; 2912, \; 5042)$
$(10863, \; 10865, \; 18817)$
$(40544, \; 40546, \; 70226)$
$(151315, \; 151317, \; 262087)$
$(564718, \; 564720, \; 978122)$
$(2107559, \; 2107561, \; 3650401)$
$(7865520, \; 7865522, \; 13623482)$
$(29354523, \; 29354525, \; 50843527)$
$(109552574, \; 109552576, \; 189750626)$

The first few triples such that   $(a, \; a+3, \; c)$

$(21, \; 24, \; 39)$
$(312, \; 315, \; 543)$
$(4365, \; 4368, \; 7563)$
$(60816, \; 60819, \; 105339)$
$(847077, \; 847080, \; 1467183)$
$(11798280, \; 11798283, \; 20435223)$
$(164328861, \; 164328864, \; 284625939)$
$(2288805792, \; 2288805795, \; 3964327923)$
$(31878952245, \; 31878952248, \; 55215964983)$
$(444016525656, \; 444016525658, \; 769059181839)$
$(6184352406957, \; 6184352406960, \; 10711612580763)$
$(86136917171760, \; 86136917171763, \; 149193516948843)$
$(1199732487997701, \; 1199732487997704, \; 2077997624703039)$
$(16710117914796072, \; 16710117914796075, \; 28942773228893703)$