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

1,1

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)).

References

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

Links

Table of n, a(n) for n=1..63.

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

Example

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.

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 *)

Crossrefs

Cf. A210250.

Sequence in context: A252072 A200856 A116301 * A045082 A257414 A179668

Adjacent sequences: A219222 A219223 A219224 * A219226 A219227 A219228

Keyword

nonn

Author

Michel Lagneau, Nov 15 2012

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

Status

approved