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

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, 85680, 87360, 92400, 100320, 110400, 115920, 118560, 140400, 147840, 182160, 187680, 201600, 215280, 235200, 236640, 244800, 255360, 257040, 265200, 277200

Offset

1,1

Comments

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 *

...........................................

Links

Table of n, a(n) for n=1..45.

Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.

E. Gürel, Solution to Problem 1472, Maximal Area of Quadrilaterals, Math. Mag. 69 (1996), 149.

Kival Ngaokrajang, Illustration of initial terms

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

Example

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.

Mathematica

SMax = 277300
Do[
  Do[
    x=S^2/(u v w);
    If[u+v+w+x//OddQ, Continue[]];
    If[v+w+x<=u, Continue[]];
    r=Sqrt[v w+u x]Sqrt[u w+v x]Sqrt[u v+w x]/(4S);
    If[r//PrimeQ//Not, Continue[]];
    (*{a, b, c, d}=(u+v+w+x)/2-{u, v, w, x}; {a, b, c, d, r, S}//Sow*);
    S//Sow; Break[]; (*to generate a table, comment out this line and uncomment previous line*)
    , {u, S^2//Divisors//Select[#, S<=#^2&]&}
    , {v, S^2/u//Divisors//Select[#, S^2<=u#^3&&#<=u&]&}
    , {w, S^2/(u v)//Divisors//Select[#, S^2<=u v#^2&&#<=v&]&}
  ]
  , {S, 24, SMax, 24}
]//Reap//Last//Last
{x, r, a, b, c, d}=.;
(* Zachary Sizer, Jan 02 2025, adapted from the program for A210250 by Albert Lau *)

Crossrefs

Cf. A210250.

Sequence in context: A211750 A211760 A333670 * A157913 A181773 A052683

Adjacent sequences: A230133 A230134 A230135 * A230137 A230138 A230139

Keyword

nonn

Author

Michel Lagneau, Oct 10 2013

Extensions

Incorrect program removed and missing term 85680 and others added by Zachary Sizer, Jan 02 2025

Status

approved