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A230136
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Integer areas A of integer-sided cyclic quadrilaterals such that the circumradius is of prime length.
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5
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48, 240, 480, 1440, 1680, 2640, 5040, 6720, 7920, 10560, 12480, 13680, 18720, 21840, 28560, 31200, 32640, 34320, 36960, 44880, 48720, 53040, 63840, 71760, 77520, 87360, 92400, 100320, 115920, 147840, 187680, 201600, 215280, 236640, 244800, 257040, 277200
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OFFSET
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1,1
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COMMENTS
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Subset of A210250. The corresponding prime circumradius are 5, 13, 17, 41, 29, 61, 53, 101, 73, 89, 97, 109, 149, 313, 257, 173,...
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.
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.
The circumradius R (the radius of the circumcircle) is given by R = sqrt(ab+cd)(ac+bd)(ad+bc)/4A.
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.
It seems that the quadrilaterals are of the form (a, a, b, b).
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.
************************************************
* A * R * a * b * c * d *
************************************************
* 48 * 5 * 6 * 6 * 8 * 8 *
* 240 * 13 * 10 * 10 * 24 * 24 *
* 480 * 17 * 16 * 16 * 30 * 30 *
* 1440 * 41 * 18 * 18 * 80 * 80 *
* 1680 * 29 * 24 * 24 * 42 * 42 *
* 2640 * 61 * 22 * 22 * 120 * 120 *
* 5040 * 53 * 56 * 56 * 90 * 90 *
* 7920 * 101 * 40 * 40 * 198 * 198 *
* 10560 * 73 * 96 * 96 * 110 * 110 *
* 12480 * 89 * 78 * 78 * 160 * 160 *
* 18720 * 97 * 130 * 130 * 144 * 144 *
...........................................
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LINKS
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EXAMPLE
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48 is in the sequence because, for (a,b,c,d) = (6,6,8,8) and :
s = (6+6+8+8)/2 = 14;
A = sqrt((14-6)(14-6)(14-8)(14-8))=48;
R = sqrt((6*6+8*8)(6*8+6*8)(6*8+6*8))/(4*48) = 960/192 = 5 is prime.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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