A210250 -id:A210250 - cp's OEIS frontend

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

Michel Lagneau, Oct 10 2013

nonn

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  *
...........................................

			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.

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

Cf. A210250.

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 *)

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

A218431 Cyclic quadrilateral numbers: numbers m = abc*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

Original entry on oeis.org

2304, 36864, 57600, 186624, 230400, 451584, 589824, 630000, 806400, 921600, 1440000, 2073600, 2822400, 2985984, 3686400, 4665600, 5531904, 6969600, 7225344, 8960000, 9437184, 10080000, 12672000, 12902400, 14745600, 15116544, 16257024, 18662400, 19079424

Offset: 1

Michel Lagneau, Oct 28 2012

A cyclic quadrilateral is a quadrilateral for which a circle can be circumscribed so that it touches each polygon vertex.
A cyclic quadrilateral number m is an integer with at least one decomposition m = a*b*c*d such that the area of the quadrilateral of sides (a,b,c,d) and the circumradius are integers. Because this property is not always unique, we introduce the notion of "cyclic quadrilateral order" for each cyclic quadrilateral number m, denoted by CQO(m). For example, CQO(2304) = 1 because the decomposition 2304 = 8*8*6*6 is unique with the quadrilateral (8,8,6,6) whose area A is given by Brahmagupta's formula: A = sqrt((s - a)*(s - b)*(s - c)*(s - d)) where the semiperimeter is s = (a+b+c+d)/2 and the circumradius R (the radius of the circumcircle) is given by R = sqrt((ab+cd)*(ac+bd)*(ad+bc))/(4A) => A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48, and R = 5, but CQO(2822400) = 2 because 2822400 = 24*24*70*70 = 40*40*42*42 and the area of the quadrilateral (24,24,70,70) equals 1680 with R = 37 and the area of the quadrilateral (40,40,42,42) also equals 1680 with R = 29.
The number of ways to write m = a*b*c*d with 1 <= a <= b <= c <= d <= m is given by A218320, thus: CQO(m) <= A218320(m).
If m is in this sequence, so is m*k^4 for any k > 0. Thus this sequence is infinite.
In view of the preceding comment, one might call "primitive" the terms m of the sequence for which there is no k > 1 such that m/k^4 is again a term of the sequence. These terms are 2304, 57600, 230400, 451584, 630000, ...

			2304 is in the sequence because 2304 = 8*8*6*6 and we obtain:
s = (8+8+6+6)/2 = 14;
A = sqrt((14-8)*(14-8)*(14-6)*(14-6)) = 48;
R = sqrt((8*8 + 6*6)*(8*6 + 8*6)*(8*6 + 8*6))/(4*48) = 5.

E. Gürel, Solution to Problem 1472, Maximal Area of Quadrilaterals, Math. Mag. 69 (1996), 149.
Eric W. Weisstein, MathWorld: Cyclic Quadrilateral.

Cf. A210250.

nn=200; lst={}; Do[s=(a+b+c+d)/2; If[IntegerQ[s], area2=(s-a)*(s-b)*(s-c)*(s-d); If[0

Typos in comment fixed by Zak Seidov and M. F. Hasler, Sep 21 2013, Sep 21 2013

A219225 Area A of the cyclic quadrilaterals PQRS with PQ>=QR>=RS>=SP, such that A, the sides, the radius of the circumcircle and the two diagonals are integers.

Original entry on oeis.org

768, 936, 1200, 2856, 3072, 3744, 4536, 4800, 5016, 5376, 6696, 6912, 7056, 7560, 7752, 8184, 8424, 9240, 10800, 11424, 11544, 12288, 12480, 12936, 14976, 16848, 18144, 18696, 19200, 19200, 20064, 21504, 23040, 23400, 24024, 25080, 25704, 25944, 26784, 27048, 27648, 27648, 27648, 27864, 28224, 28560, 30000, 30240, 31008, 32736, 33696, 34560, 36960, 36960, 37632, 40392, 40560, 40824, 41064, 41184, 42240, 42840, 43200

Offset: 1

Michel Lagneau, Nov 15 2012

nonn

Subsequence of A210250.
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 diagonals of a cyclic quadrilateral have length:
p = sqrt((ab+cd)(ac+bd)/(ad+bc))
q = sqrt((ac+bd)(ad+bc)/(ab+cd)).

			936 is in the sequence because, with sides (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;
p = sqrt((14*30+40*48)( 14*40+30*48)/( 14*48+30*40)) = 50;
q= sqrt((14*40+30*48)( 14*48+30*40)/( 14*30+40*48))  = 40.

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32.

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.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

Cf. A210250.

SMax=10000;
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//IntegerQ//Not, Continue[]];
    {a, b, c, d}=(u+v+w+x)/2-{u, v, w, x};
    If[4S r/(a b+c d)//IntegerQ//Not,Continue[]];
    If[4S r/(a d+b c)//IntegerQ//Not,Continue[]];
    (*{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}=.; (* Albert Lau, May 25 2016 *)

Incorrect Mathematica program removed by Albert Lau, May 25 2016

Missing terms 18144, 20064, 21504 and more term from Albert Lau, May 25 2016

A228383 Area A of the triangle such that A, the sides, and the inradius are integers.

Original entry on oeis.org

6, 24, 30, 36, 42, 48, 54, 60, 66, 84, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 210, 216, 240, 252, 264, 270, 294, 300, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528, 540, 546, 570

Offset: 1

Michel Lagneau, Aug 21 2013

nonn

The sequences A208984 and A185210 are subsequences of this sequence. The corresponding inradius r are 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 3, 4, 3, ...
The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. The inradius r is given by r = A/s.
a(n) is divisible by 6 and the squares of the form 36k^2 are in the sequence.

			24 is in the sequence because for (a, b, c) = (6, 8, 10) => s =(6 + 8 + 10)/2 = 12; A = sqrt(12*(12-6)*(12-8)*(12-10)) = sqrt(576)= 24; r = A/s = 2.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), pp. 32-34.
Eric W. Weisstein, MathWorld: Inradius

Cf. A188158, A120572, A210250, A208984, A185210.

nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]] && IntegerQ[Sqrt[area2]/s], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

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

Original entry on oeis.org

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

Michel Lagneau, Dec 07 2013

nonn

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 |
.......................................

			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.

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.

Wolfram MathWorld, Cyclic Quadrilateral

Cf. A210250.

nn=500;lst={};Do[s=(a+b+c+d)/2;If[IntegerQ[s],area2=(s-a)*(s-b)*(s-c)*(s-d);If[0

A242778 Sides (a,c) of cyclic quadrilaterals of integer sides (a,b,c,d), integer areas, and integer circumradius such that a=b and c=d.

Original entry on oeis.org

6, 8, 10, 24, 12, 16, 14, 48, 16, 30, 18, 24, 18, 80, 20, 48, 22, 120, 24, 32, 24, 70, 26, 168, 28, 96, 30, 40, 30, 72, 30, 224, 32, 60, 32, 126, 34, 288, 36, 48, 36, 160, 38, 360, 40, 42, 40, 96, 40, 198, 42, 56, 42, 144, 42, 440, 44, 240, 46, 528, 48, 64

Offset: 1

Michel Lagneau, May 22 2014

nonn

In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribing circle, and the vertices are said to be concyclic.
The area A of any 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) of any cyclic quadrilateral is given by
R = sqrt((ab+cd)(ac+bd)(ad+bc))/(4A).
Many cyclic quadrilaterals [a, b, c, d] with integer sidelengths, integer area, and integer circumradius have the property that a = b and c = d, thus forming a kite with two right angles, with the long diagonal of the kite being a diameter of the circle; thus the circumradius is R = sqrt(a^2 + c^2)/2.  Since Brahmagupta's formula is invariant upon permutation of the sides, the area of such a kite is the same as that of the rectangle with sides [a, c, b, d]. So in this case s = a+c, and A = a*c. In particular, the double of any Pythagorean triple will satisfy our requirements.
Nevertheless, there also exist cyclic quadrilaterals with integer sidelengths, integer area, and integer circumradius, whose four sides are distinct; for example, [a, b, c, d] = [ 14, 30, 40, 48] => A = 936 and R = 25.

			(a(1),a(2)) = (6,8) because, for (a,b,c,d) = (6,6,8,8) we obtain:
s = a + c = 6+8 = 14;
A = a*c = 6*8 = 48;
R = sqrt(a^2 + c^2)/2 = sqrt(6^2 + 8^2)/2 = 5.

Eric Weisstein's World of Mathematics, Cyclic Quadrilateral

Cf. A210250, A297790, A298860, A298907.

nn=1500;lst={};Do[s=(2*a+2*c)/2;If[IntegerQ[s],area2=(s-a)^2*(s-c)^2;If[0

Definition and comments extended and/or corrected by Gregory Gerard Wojnar, Nov 10 2018

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

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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A218431 Cyclic quadrilateral numbers: numbers m = a*b*c*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.

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A219225 Area A of the cyclic quadrilaterals PQRS with PQ>=QR>=RS>=SP, such that A, the sides, the radius of the circumcircle and the two diagonals are integers.

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A228383 Area A of the triangle such that A, the sides, and the inradius are integers.

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

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A242778 Sides (a,c) of cyclic quadrilaterals of integer sides (a,b,c,d), integer areas, and integer circumradius such that a=b and c=d.

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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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A218431 Cyclic quadrilateral numbers: numbers m = abc*d such that the integers a,b,c,d are the sides of a cyclic quadrilateral whose area and circumradius are integers.