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A230136 Integer areas A of integer-sided cyclic quadrilaterals such that the circumradius is of prime length.

%I A230136 #25 Feb 16 2025 08:33:20
%S A230136 48,240,480,1440,1680,2640,5040,6720,7920,10560,12480,13680,18720,
%T A230136 21840,28560,31200,32640,34320,36960,44880,48720,53040,63840,71760,
%U A230136 77520,85680,87360,92400,100320,110400,115920,118560,140400,147840,182160,187680,201600,215280,235200,236640,244800,255360,257040,265200,277200
%N A230136 Integer areas A of integer-sided cyclic quadrilaterals such that the circumradius is of prime length.
%C A230136 Subset of A210250. The corresponding prime circumradius are 5, 13, 17, 41, 29, 61, 53, 101, 73, 89, 97, 109, 149, 313, 257, 173,...
%C A230136 In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed, and the vertices are said to be concyclic.
%C A230136 The area A of a cyclic quadrilateral with sides a, b, c, d is given by Brahmagupta’s formula : A = sqrt((s - a)(s -b)(s - c)(s - d))  where s, the semiperimeter is s= (a+b+c+d)/2.
%C A230136 The circumradius R (the radius of the circumcircle) is given by R = sqrt(ab+cd)(ac+bd)(ad+bc)/4A.
%C A230136 The corresponding R of a(n) are not unique, for example for a(5) = 1680 => (a,b,c,d) = (24, 24, 70, 70) with R = 37 and (a,b,c,d) = (40, 40, 42, 42) with R = 29.
%C A230136 It seems that the quadrilaterals are of the form (a, a, b, b).
%C A230136 The following table gives the first values (A, R, a, b, c, d) where A is the integer area, R the radius of the circumcircle, and a, b, c, d are  the integer sides of the cyclic quadrilateral.
%C A230136 ************************************************
%C A230136 *    A  *     R   *   a  *   b  *   c  *    d  *
%C A230136 ************************************************
%C A230136 *    48  *    5   *   6  *   6  *   8   *   8  *
%C A230136 *   240  *   13   *  10  *  10  *  24   *  24  *
%C A230136 *   480  *   17   *  16  *  16  *  30   *  30  *
%C A230136 *  1440  *   41   *  18  *  18  *  80   *  80  *
%C A230136 *  1680  *   29   *  24  *  24  *  42   *  42  *
%C A230136 *  2640  *   61   *  22  *  22  * 120   * 120  *
%C A230136 *  5040  *   53   *  56  *  56  *  90   *  90  *
%C A230136 *  7920  *  101   *  40  *  40  * 198   * 198  *
%C A230136 * 10560  *   73   *  96  *  96  * 110   * 110  *
%C A230136 * 12480  *   89   *  78  *  78  * 160   * 160  *
%C A230136 * 18720  *   97   * 130  * 130  * 144   * 144  *
%C A230136 ...........................................
%H A230136 Mohammad K. Azarian, <a href="http://www.jstor.org/stable/25678790">Solution to Problem S125: Circumradius and Inradius</a>, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
%H A230136 E. Gürel, <a href="http://www.jstor.org/stable/2690677?seq=7">Solution to Problem 1472, Maximal Area of Quadrilaterals</a>, Math. Mag. 69 (1996), 149.
%H A230136 Kival Ngaokrajang, <a href="/A230136/a230136.pdf">Illustration of initial terms</a>
%H A230136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>
%e A230136 48 is in the sequence because, for (a,b,c,d) = (6,6,8,8) and :
%e A230136 s = (6+6+8+8)/2 = 14;
%e A230136 A = sqrt((14-6)(14-6)(14-8)(14-8))=48;
%e A230136 R = sqrt((6*6+8*8)(6*8+6*8)(6*8+6*8))/(4*48) = 960/192 = 5 is prime.
%t A230136 SMax = 277300
%t A230136 Do[
%t A230136   Do[
%t A230136     x=S^2/(u v w);
%t A230136     If[u+v+w+x//OddQ, Continue[]];
%t A230136     If[v+w+x<=u, Continue[]];
%t A230136     r=Sqrt[v w+u x]Sqrt[u w+v x]Sqrt[u v+w x]/(4S);
%t A230136     If[r//PrimeQ//Not, Continue[]];
%t A230136     (*{a, b, c, d}=(u+v+w+x)/2-{u, v, w, x}; {a, b, c, d, r, S}//Sow*);
%t A230136     S//Sow; Break[]; (*to generate a table, comment out this line and uncomment previous line*)
%t A230136     , {u, S^2//Divisors//Select[#, S<=#^2&]&}
%t A230136     , {v, S^2/u//Divisors//Select[#, S^2<=u#^3&&#<=u&]&}
%t A230136     , {w, S^2/(u v)//Divisors//Select[#, S^2<=u v#^2&&#<=v&]&}
%t A230136   ]
%t A230136   , {S, 24, SMax, 24}
%t A230136 ]//Reap//Last//Last
%t A230136 {x, r, a, b, c, d}=.;
%t A230136 (* _Zachary Sizer_, Jan 02 2025, adapted from the program for A210250 by _Albert Lau_ *)
%Y A230136 Cf. A210250.
%K A230136 nonn
%O A230136 1,1
%A A230136 _Michel Lagneau_, Oct 10 2013
%E A230136 Incorrect program removed and missing term 85680 and others added by _Zachary Sizer_, Jan 02 2025