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
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
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