A233315 -id:A233315 - cp's OEIS frontend

A297790 Areas of integer side-length cyclic quadrilaterals, excluding rectangles and kites with two right angles, with integer area.

12, 15, 16, 18, 20, 21, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 64, 65, 66, 70, 72, 75, 78, 80, 84, 85, 88, 90, 95, 96, 99, 100, 102, 104, 105, 108, 110, 112, 115, 117, 120, 126, 128, 130, 132, 135, 136, 140, 144, 150, 152, 153, 154, 156, 160, 162, 165, 168, 170, 171, 175, 176, 180, 182, 184, 189, 190, 192, 195, 196, 198, 200, 203, 204, 207, 208, 210, 216, 217, 220, 224, 225, 228, 230, 231, 232, 234, 238

Offset: 1

Gregory Gerard Wojnar, Jan 28 2018

nonn

Daniel J Hardisky, Omission of Area 64

Cf. A298907, A298860, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.

Name clarified by Gregory Gerard Wojnar, Oct 31 2018

Missing terms 64, 112, and 162 inserted by Daniel J Hardisky, Apr 24 2025

A273890 Integer area A of the cyclic quadrilaterals such that A, the sides and the two diagonals are integers.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Jun 02 2016

nonn

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’s 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  |
..................................................................

			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.

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

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

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

A298860 Primitive cyclic quadrilaterals with integer area.

Original entry on oeis.org

1, 3, 6, 8, 18, 12, 1, 5, 5, 7, 18, 16, 1, 2, 8, 9, 20, 12, 1, 5, 5, 9, 20, 15, 1, 4, 7, 8, 20, 18, 2, 5, 5, 8, 20, 20, 2, 5, 5, 10, 22, 18, 3, 5, 5, 9, 22, 24, 2, 4, 7, 11, 24, 20, 3, 5, 5, 11, 24, 21, 4, 5, 5, 10, 24, 28, 2, 6, 7, 9, 24, 30, 4, 5, 5, 12, 26, 24, 3, 4, 8, 11, 26, 30, 4, 5, 7, 10, 26, 36, 2, 5, 10, 11, 28, 36, 1, 7, 8, 14, 30, 28, 1, 8, 9, 12, 30, 42

Offset: 1

Gregory Gerard Wojnar, Jan 27 2018

Entries are listed as sextuples: (a,b,c,d), Perimeter, Area. They are ordered first by perimeter, second by area, third by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.

We observe that the number of odd integers in any quadruple is always an even number.

			The first row of the table gives sidelengths (a,b,c,d)=(1,3,6,8) with perimeter=18 and area=12. Thus:
  a b c  d Perim Area
  = = = == ===== ====
  1 3 6  8   18   12
  1 5 5  7   18   16
  1 2 8  9   20   12
  1 5 5  9   20   15
  1 4 7  8   20   18
  2 5 5  8   20   20
  2 5 5 10   22   18
  3 5 5  9   22   24
  2 4 7 11   24   20
  3 5 5 11   24   21
  4 5 5 10   24   28
  etc.

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Wikipedia, Cyclic quadrilateral

Cf. A298907, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.

A298907 Primitive cyclic quadrilaterals with integer area.

Original entry on oeis.org

1, 3, 6, 8, 1, 5, 5, 7, 1, 2, 8, 9, 1, 5, 5, 9, 1, 4, 7, 8, 2, 5, 5, 8, 2, 5, 5, 10, 3, 5, 5, 9, 2, 4, 7, 11, 3, 5, 5, 11, 4, 5, 5, 10, 2, 6, 7, 9, 4, 5, 5, 12, 3, 4, 8, 11, 4, 5, 7, 10, 2, 5, 10, 11, 1, 7, 8, 14, 1, 8, 9, 12, 3, 7, 9, 11, 1, 6, 10, 15, 2, 7, 9, 14, 1, 7, 11, 13, 6, 7, 8, 11, 1, 10, 10, 13, 2, 9, 11, 12, 3, 6, 13, 14, 3, 8, 10, 15

Offset: 1

Gregory Gerard Wojnar, Jan 28 2018

Entries are listed as quadruples: (a,b,c,d). They are ordered first by perimeter, second by area, then by a, then b, then c, then d. Rectangles and kites with two right angles are not listed; thus a < b <= c <= d. By "primitive" we mean (a,b,c,d) is not a multiple of any earlier quadruple.

It appears that the number of odd sidelengths in any quadruple is always 0, 2, or 4.

			We list here the early quadruplets, in parentheses, augmented by the associated perimeter and area to justify the ordering of the quadruplets:
(a,  b,  c,  d)  Perim  Area
===============  =====  ====
(1,  3,  6,  8)    18    12
(1,  5,  5,  7)    18    16
(1,  2,  8,  9)    20    12
(1,  5,  5,  9)    20    15
(1,  4,  7,  8)    20    18
(2,  5,  5,  8)    20    20
(2,  5,  5, 10)    22    18
(3,  5,  5,  9)    22    24
(2,  4,  7, 11)    24    20
(3,  5,  5, 11)    24    21
(4,  5,  5, 10)    24    28
(2,  6,  7,  9)    24    30
etc.

Cf. A298860, A297790, A210250, A230136, A131020, A218431, A219225, A233315, A242778, A273691, A273890.

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A297790 Areas of integer side-length cyclic quadrilaterals, excluding rectangles and kites with two right angles, with integer area.

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A273890 Integer area A of the cyclic quadrilaterals such that A, the sides and the two diagonals are integers.

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A298860 Primitive cyclic quadrilaterals with integer area.

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A298907 Primitive cyclic quadrilaterals with integer area.

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