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A273890 Integer area A of the cyclic quadrilaterals such that A, the sides and the two diagonals are integers. 3
192, 234, 300, 432, 714, 768, 936, 1134, 1200, 1254, 1344, 1674, 1728, 1764, 1890, 1938, 2046, 2106, 2226, 2310, 2352, 2700, 2856, 2886, 3072, 3120, 3234, 3744, 3888, 3990, 4092, 4212, 4368, 4536, 4674, 4800, 4914, 5016, 5292, 5376, 5760, 5850, 6006, 6270, 6426 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The areas of the primitive cyclic quadrilaterals of this sequence are in A273691.

This sequence contains A233315 (768, 936, 1200,...).

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‚Äôs formula : A = sqrt((s - a)(s -b)(s - c)(s - d))  where s, the semiperimeter is s= (a+b+c+d)/2.

In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as

p = sqrt((ac+bd)(ad+bc)/(ab+cd)) and q = sqrt((ac+bd)(ab+cd)/(ad+bc)).

The circumradius R (the radius of the circumcircle) is given by :

R = sqrt((ab+cd)(ac+bd)(ad+bc))/4A.

The corresponding sides of a(n) are not unique, for example for a(6) = 768 => (a,b,c,d) = (25, 25, 25, 39) or (a,b,c,d) = (14, 30, 30, 50).

The following table gives the first values (A, a, b, c, d, p, q, R) where A is the integer area, a, b, c, d are the integer sides of the cyclic quadrilateral, p, q are the integer diagonals, and R .

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

|    A   |   a   |   b   |   c   |   d    |   p   |  q   |  R    |

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

|   192  |   7   |   15  |   15  |   25   |   20  |  24  | 25/2  |

|   234  |   7   |   15  |   20  |   24   |   20  |  25  | 25/2  |

|   300  |  15   |   15  |   20  |   20   |   24  |  25  | 25/2  |

|   432  |  11   |   25  |   25  |   25   |   30  |  30  | 125/8 |

|   714  |  16   |   25  |   33  |   60   |   39  |  52  | 65/2  |

|   768  |  25   |   25  |   25  |   39   |   40  |  40  | 125/6 |

|   768  |  14   |   30  |   30  |   50   |   40  |  48  | 25    |

|   936  |  14   |   30  |   40  |   48   |   40  |  50  | 25    |

|  1134  |  16   |   25  |   52  |   65   |   39  |  63  | 65/2  |

|  1200  |  30   |   30  |   40  |   40   |   48  |  50  | 25    |

|  1254  |  16   |   25  |   60  |   63   |   39  |  65  | 65/2  |

|  1344  |  25   |   33  |   39  |   65   |   52  |  60  | 65/2  |

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

LINKS

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

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

EXAMPLE

192 is in the sequence because, for (a,b,c,d) = (7,15,15,25) we find:

s = (7+15+15+25)/2 = 31;

A = sqrt((31-7)(31-15)(31-15)(31-25)) = 192;

p = sqrt((7*15+15*25)*(7*25+15*15)/(7*15+15*25)) = 20;

q = sqrt((7*15+15*25)*(7*15+15*25)/(7*25+15*15)) = 24.

MATHEMATICA

nn=200; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); d1=Sqrt[(a*c+b*d)*(a*d+b*c)/(a*b+c*d)]; d2=Sqrt[(a*c+b*d)*(a*b+c*d)/(a*d+b*c)]; If[0<area2 && IntegerQ[Sqrt[area2]] && IntegerQ[d1]&& IntegerQ[d2], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}, {d, c}]; Union[lst]

CROSSREFS

Cf. A210250, A218431, A219225, A230136, A233315, A242778, A273691.

Sequence in context: A289219 A190892 A039667 * A045077 A234135 A030632

Adjacent sequences:  A273887 A273888 A273889 * A273891 A273892 A273893

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 02 2016

STATUS

approved

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Last modified May 26 16:52 EDT 2020. Contains 334627 sequences. (Running on oeis4.)