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%I A273691 #5 May 28 2016 07:29:54
%S A273691 12,60,108,120,120,168,192,192,234,240,300,360,360,420,420,420,420,
%T A273691 420,420,432,540,540,588,600,660,660,714,768,840,924,960,960,966,1008,
%U A273691 1008,1008,1080,1080,1080,1092,1134,1200
%N A273691 Integer area of primitive cyclic quadrilaterals with integer sides and rational radius.
%C A273691 Given 4 segments a,b,c,d, there is a unique circumcircle such that these segments can be placed inside to form cyclic quadrilaterals. There are 3 ways to place these segments: abcd,acbd,adbc.
%C A273691 Primitive means a,b,c,d share no common factor.
%C A273691 The area S = sqrt[(s-a)(s-b)(s-c)(s-d)] where s=(a+b+c+d)/2 is the semiperimeter.
%C A273691 The circumradius R=Sqrt[a b+c d]*Sqrt[a c+b d]*Sqrt[a d+b c]/(4S)
%C A273691 The length of the diagonal separating a-b and c-d is (4S R)/(a b+c d), the other diagonal can be obtain by swapping b,c or swapping b,d.
%C A273691 It follows that if the sides and area are integers, then (any diagonal is rational) <=> (circumradius is rational) <=> (all diagonals are rational).
%C A273691 From empirical observation, the area seems to be a multiple of 6. (If so, the program could be modified to run 6 times as fast.)
%e A273691 a,  b,  c,  d,  S,   r
%e A273691 4,  4,  3,  3,  12,  5/2
%e A273691 12, 12, 5,  5,  60,  13/2
%e A273691 14, 13, 13, 4,  108, 65/8
%e A273691 15, 15, 8,  8,  120, 17/2
%e A273691 21, 10, 10, 9,  120, 85/8
%e A273691 24, 24, 7,  7,  168, 25/2
%e A273691 21, 13, 13, 11, 192, 65/6
%e A273691 25, 15, 15, 7,  192, 25/2
%e A273691 24, 20, 15, 7,  234, 25/2
%t A273691 SMax=1200;
%t A273691 Do[
%t A273691   x=S^2/(u v w);
%t A273691   If[u+v+w+x//OddQ,Continue[]];
%t A273691   If[v+w+x<=u,Continue[]];
%t A273691   {a,b,c,d}=(u+v+w+x)/2-{x,w,v,u};
%t A273691   If[GCD[a,b,c,d]>1,Continue[]];
%t A273691   R=(Sqrt[v w+u x]Sqrt[u w+v x]Sqrt[u v+w x])/(4S);
%t A273691   If[R\[NotElement]Rationals,Continue[]];
%t A273691   S(*{a,b,c,d,"",S,R,"",(4S R)/(a d+b c),(4S R)/(a c+b d),(4S R)/(a b+c d)}*)//Sow;
%t A273691   ,{S,1(*6*),SMax,1(*6*)}(*assuming S mod 6 = 0, set to 6 to run faster*)
%t A273691   ,{u,S^2//Divisors//Select[#,S<=#^2&&#<=S&]&}
%t A273691   ,{v,S^2/u//Divisors//Select[#,S^2<=u#^3&&u/3<#<=u&]&}
%t A273691   ,{w,S^2/(u v)//Divisors//Select[#,S^2<=u v#^2&&(u-v)/2<#<=v&]&}
%t A273691 ]//Reap//Last//Last(*//TableForm*)
%t A273691 {S,R,x,a,b,c,d}=.;
%K A273691 nonn
%O A273691 1,1
%A A273691 _Albert Lau_, May 28 2016

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