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

Author

Michel Lagneau, Dec 07 2013

Keywords

• nonn

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 |

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

Examples

			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.
		

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

• Wolfram MathWorld, Cyclic Quadrilateral

Crossrefs

Cf. A210250.

Programs

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