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.

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

Author

Michel Lagneau, Nov 15 2012

Keywords

• nonn

Comments

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)).

Examples

			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.
		

References

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

Links

• 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

Crossrefs

Cf. A210250.

Programs

Mathematica

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 *)

Extensions

Incorrect Mathematica program removed by Albert Lau, May 25 2016

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