A233315 Integer areas A of integer-sided cyclic quadrilaterals such that the length of the circumradius is a perfect square.

672, 768, 936, 1200, 10752, 12288, 14976, 19200, 34560, 40560, 48840, 54432, 57120, 62208, 75816, 97200, 138720, 154560, 172032, 196608, 239616, 307200, 420000, 480000, 552960, 585000, 648960, 750000, 781440, 870912, 913920, 995328, 1213056, 1555200, 2219520

Offset

1,1

Comments

Subset of A210250. The corresponding square circumradius are 25,25,25,25,100,100,100,100,169,169,169,169,225, ...

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 areas A of the primitive quadrilaterals of sides (a,b,c,d) are 672,768,936,1200,34560,40560,48840,57120,...

The areas of the non-primitive quadrilaterals of sides (a*p^2, b*p^2, c*p^2, d*p^2) are in the sequence with the value A*p^4.

The following table gives the first values (A, a, b, c, d, R) 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 | a | b | c | d | R |

+-------+-----+-----+-----+-----+-----+

| 672 | 14 | 14 | 48 | 48 | 25 |

| 768 | 14 | 30 | 30 | 50 | 25 |

| 936 | 14 | 30 | 40 | 48 | 25 |

| 1200 | 30 | 30 | 40 | 40 | 25 |

| 10752 | 56 | 56 | 192 | 192 | 100 |

| 12288 | 56 | 120 | 120 | 200 | 100 |

| 14976 | 56 | 120 | 160 | 192 | 100 |

| 19200 | 120 | 120 | 160 | 160 | 100 |

| 34560 | 130 | 130 | 238 | 338 | 169 |

| 40560 | 130 | 130 | 312 | 312 | 169 |

| 48840 | 130 | 238 | 240 | 312 | 169 |

| 54432 | 126 | 126 | 432 | 432 | 225 |

| 57120 | 238 | 238 | 240 | 240 | 169 |

| 62208 | 126 | 270 | 270 | 450 | 225 |

| 75816 | 126 | 270 | 360 | 432 | 225 |

| 97200 | 270 | 270 | 360 | 360 | 225 |

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

References

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Links

Table of n, a(n) for n=1..35.

Wolfram MathWorld, Cyclic Quadrilateral

Example

936 is in the sequence because, for (a,b,c,d) = (14,30,40,48) we obtain:
s = (14+30+40+48)/2 = 66;
A = sqrt((66-14)*(66-30)*(66-40)*(66-48))=936;
R = sqrt((14*30+40*48)*(14*40+30*48)*(14*48+30*40))/(4*936) = 93600/3744 = 25 is square.

Mathematica

nn=500; 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]]&&IntegerQ[Sqrt[Sqrt[(a*b+c*d)*(a*c+b*d)*(a*d+b*c)/area2]/4]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]

Crossrefs

Cf. A210250.

Sequence in context: A281526 A053085 A057695 * A157363 A308574 A234732

Adjacent sequences: A233312 A233313 A233314 * A233316 A233317 A233318

Keyword

nonn

Author

Michel Lagneau, Dec 07 2013

Status

approved