Editing Pythagorean triple (section)

===Heronian triangle triples===
{{main|Heronian triangle}}

A '''Heronian triangle''' is commonly defined as one with integer sides whose area is also an integer. The lengths of the sides of such a triangle form a '''Heronian triple''' {{math|(''a, b, c'')}} for {{math|''a'' ≤ ''b'' ≤ ''c''}}.
Every Pythagorean triple is a Heronian triple, because at least one of the legs {{math|''a''}}, {{math|''b''}} must be even in a Pythagorean triple, so the area ''ab''/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example {{math|(4, 13, 15)}} with area 24 shows.

If {{math|(''a'', ''b'', ''c'')}} is a Heronian triple, so is {{math|(''ka'', ''kb'', ''kc'')}} where {{math|''k''}} is any positive integer; its area will be the integer that is {{math|''k''{{sup|2}}}} times the integer area of the {{math|(''a'', ''b'', ''c'')}} triangle.
The Heronian triple {{math|(''a'', ''b'', ''c'')}} is '''primitive''' provided ''a'', ''b'', ''c'' are [[coprime integers#Coprimality in sets|setwise coprime]].  (With primitive Pythagorean triples the stronger statement that they are ''pairwise'' coprime also applies, but with primitive Heronian triangles the stronger statement does not always hold true, such as with {{math|(7, 15, 20)}}.) Here are a few of the simplest primitive Heronian triples that are not Pythagorean triples:

: (4, 13, 15) with area 24
: (3, 25, 26) with area 36
: (7, 15, 20) with area 42
: (6, 25, 29) with area 60
: (11, 13, 20) with area 66
: (13, 14, 15) with area 84
: (13, 20, 21) with area 126

By [[Heron's formula]], the extra condition for a triple of positive integers {{math|(''a'', ''b'', ''c'')}} with {{math|''a'' < ''b'' < ''c''}} to be Heronian is that

: {{math|(''a''{{sup|2}} + ''b''{{sup|2}} + ''c''{{sup|2}}){{sup|2}} − 2(''a''{{sup|4}} + ''b''{{sup|4}} + ''c''{{sup|4}})}}
or equivalently
: {{math|2(''a''{{sup|2}}''b''{{sup|2}} + ''a''{{sup|2}}''c''{{sup|2}} + ''b''{{sup|2}}''c''{{sup|2}}) − (''a''{{sup|4}} + ''b''{{sup|4}} + ''c''{{sup|4}})}}

be a nonzero perfect square divisible by 16.