A230136 - OEIS

%I #15 Feb 25 2023 15:17:47

%S 48,240,480,1440,1680,2640,5040,6720,7920,10560,12480,13680,18720,

%T 21840,28560,31200,32640,34320,36960,44880,48720,53040,63840,71760,

%U 77520,87360,92400,100320,115920,147840,187680,201600,215280,236640,244800,257040,277200

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

%C 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 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 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 The circumradius R (the radius of the circumcircle) is given by R = sqrt(ab+cd)(ac+bd)(ad+bc)/4A.

%C 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 It seems that the quadrilaterals are of the form (a, a, b, b).

%C 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 ************************************************

%C *    A  *     R   *   a  *   b  *   c  *    d  *

%C ************************************************

%C *    48  *    5   *   6  *   6  *   8   *   8  *

%C *   240  *   13   *  10  *  10  *  24   *  24  *

%C *   480  *   17   *  16  *  16  *  30   *  30  *

%C *  1440  *   41   *  18  *  18  *  80   *  80  *

%C *  1680  *   29   *  24  *  24  *  42   *  42  *

%C *  2640  *   61   *  22  *  22  * 120   * 120  *

%C *  5040  *   53   *  56  *  56  *  90   *  90  *

%C *  7920  *  101   *  40  *  40  * 198   * 198  *

%C * 10560  *   73   *  96  *  96  * 110   * 110  *

%C * 12480  *   89   *  78  *  78  * 160   * 160  *

%C * 18720  *   97   * 130  * 130  * 144   * 144  *

%C ...........................................

%H 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 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 Kival Ngaokrajang, <a href="/A230136/a230136.pdf">Illustration of initial terms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>

%e 48 is in the sequence because, for (a,b,c,d) = (6,6,8,8) and :

%e s = (6+6+8+8)/2 = 14;

%e A = sqrt((14-6)(14-6)(14-8)(14-8))=48;

%e R = sqrt((6*6+8*8)(6*8+6*8)(6*8+6*8))/(4*48) = 960/192 = 5 is prime.

%t nn=1000; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); If[0 < area2 && IntegerQ[Sqrt[area2]] && PrimeQ[Sqrt[(a*b+c*d)*(a*c+b*d)*(a*d+b*c)/((s-a)*(s-b)*(s-c)*(s-d))]/4], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]

%Y Cf. A210250.

%K nonn

%O 1,1

%A _Michel Lagneau_, Oct 10 2013