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A298860 Primitive cyclic quadrilaterals with integer area.

%I A298860 #22 Feb 16 2025 08:33:53
%S A298860 1,3,6,8,18,12,1,5,5,7,18,16,1,2,8,9,20,12,1,5,5,9,20,15,1,4,7,8,20,
%T A298860 18,2,5,5,8,20,20,2,5,5,10,22,18,3,5,5,9,22,24,2,4,7,11,24,20,3,5,5,
%U A298860 11,24,21,4,5,5,10,24,28,2,6,7,9,24,30,4,5,5,12,26,24,3,4,8,11,26,30,4,5,7,10,26,36,2,5,10,11,28,36,1,7,8,14,30,28,1,8,9,12,30,42
%N A298860 Primitive cyclic quadrilaterals with integer area.
%C A298860 Entries are listed as sextuples: (a,b,c,d), Perimeter, Area. They are ordered first by perimeter, second by area, third by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.
%C A298860 We observe that the number of odd integers in any quadruple is always an even number.
%H A298860 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>
%H A298860 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclic_quadrilateral">Cyclic quadrilateral</a>
%e A298860 The first row of the table gives sidelengths (a,b,c,d)=(1,3,6,8) with perimeter=18 and area=12. Thus:
%e A298860   a b c  d Perim Area
%e A298860   = = = == ===== ====
%e A298860   1 3 6  8   18   12
%e A298860   1 5 5  7   18   16
%e A298860   1 2 8  9   20   12
%e A298860   1 5 5  9   20   15
%e A298860   1 4 7  8   20   18
%e A298860   2 5 5  8   20   20
%e A298860   2 5 5 10   22   18
%e A298860   3 5 5  9   22   24
%e A298860   2 4 7 11   24   20
%e A298860   3 5 5 11   24   21
%e A298860   4 5 5 10   24   28
%e A298860   etc.
%Y A298860 Cf. A298907, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.
%K A298860 nonn,tabf
%O A298860 1,2
%A A298860 _Gregory Gerard Wojnar_, Jan 27 2018