Editing Pythagorean triple (section)

==Pythagorean triangles in a 2D lattice==
A 2D [[Lattice (group)|lattice]] is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at {{math|(''x'', ''y'')}} where {{math|''x''}} and {{math|''y''}} range over all positive and negative integers. Any Pythagorean triangle with triple {{math|(''a'', ''b'', ''c'')}} can be drawn within a 2D lattice with vertices at coordinates {{math|(0, 0)}}, {{math|(''a'', 0)}} and {{math|(0, ''b'')}}. The count of lattice points lying strictly within the bounds of the triangle is given by&nbsp;&nbsp; <math>\tfrac{(a-1)(b-1)-\gcd{(a,b)}+1}{2};</math><ref>{{citation|title=Recreational Mathematics|last=Yiu|first=Paul|work=Course Notes |at=Ch. 2, p. 110 |publisher=Dept. of Mathematical Sciences, Florida Atlantic University |year=2003|url=http://math.fau.edu/Yiu/RecreationalMathematics2003.pdf}}</ref> for primitive Pythagorean triples this interior lattice count is&nbsp;&nbsp;<math>\tfrac{(a-1)(b-1)}{2}.</math> The area (by [[Pick's theorem]] equal to one less than the interior lattice count plus half the boundary lattice count) equals&nbsp;&nbsp;<math>\tfrac{ab}{2}</math>&nbsp;.

The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides {{math|(20, 21, 29), (12, 35, 37)}} and common area 210&nbsp;{{OEIS|id=A093536}}. The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with {{math|(18108, 252685, 253333), (28077, 162964, 165365)}} and interior lattice count 2287674594&nbsp;{{OEIS|id=A225760}}.  Three primitive Pythagorean triples have been found sharing the same area: {{math|(4485, 5852, 7373)}}, {{math|(3059, 8580, 9109)}}, {{math|(1380, 19019, 19069)}} with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count.