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%I #14 Aug 06 2024 11:28:39
%S 16,18,20,30
%N Areas of integer-sided cyclic quadrilaterals whose area equals their perimeter.
%C There are no further terms. Note that without the condition "integer-sided" there are other solutions, such as (1, 17/2, 17/2, 16) which has perimeter and area 34.
%H Ralph H. Buchholz and James A. MacDougall, <a href="https://citeseerx.ist.psu.edu/pdf/56dd5d57a66e0864ff6370f7ebbb0443af9e17e2">Cyclic Polygons with Rational Sides and Area</a>, 2001. <a href="https://doi.org/10.1016/j.jnt.2007.05.005">JNT 128 (2008) 17-48</a>
%e The areas or perimeters 16, 18, 20, 30 pertain respectively to cyclic quadrilaterals with sides (4, 4, 4, 4), (3, 3, 6, 6), (2, 5, 5, 8), (5, 5, 6, 14).
%t lst={}; Do[s=(a+b+c+d)/2; If[s>a, (K=Sqrt[(s-a)(s-b)(s-c)(s-d)]; If[IntegerQ[K]&&K==2s, AppendTo[lst, Sort@{a,b,c,d}]])], {a, 1, 15}, {b, 1, a}, {c, 1, b}, {d, 1, c}]; lst
%Y Cf. A098030, A290451. First four terms of A161874.
%K fini,full,nonn,changed
%O 1,1
%A _Frank M Jackson_, Oct 02 2021
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