A219193 Primitive bicentric quadrilaterals defined in A219192.

2352, 69360, 253920, 645792, 2824080, 3457440, 3711120, 13308240, 25163712, 26858160, 58552032, 112031040

Offset: 1

Author

Michel Lagneau, Nov 14 2012

Keywords

• nonn

Comments

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The sequence A219192 gives the areas A of bicentric quadrilateral with sides (a,b,c,d), inradius r and circumradius R integer. For any such A, A*k^2 is again an element of A219192 for any integer k>0. Terms in A219192 which are not of that form (with k>1) are called "primitive", and listed here.

The term A219192(2) = 9408 is not in this sequence since 9408 = 2^2*2352 and 2352 = a(1).

The term n = 253920 = 2 ^ 5 * 3 * 5 * 23 ^ 2 is divisible by 4^2 and 23^2, but neither of n/3^2, n/5^2, n/23^2 is in A219192, therefore n is in this sequence.

Crossrefs

Cf. A210250, A219192.

Programs

Maple

The Maple program gives the vector (A, a, b, c, d, r, R).
A is the area, a,b,c,d are the sides of the quadrilateral, r is the inradius and R is the circumradius.
with(numtheory):k:=1:T:=array(1..5000):T[k]:=2352:kk:=0:nn:=15000:for a from 1 to nn do: b:=a: for c from b to nn do: for d from c to c while(sqrt(a*b*c*d)=floor(sqrt(a*b*c*d))) do:s:=(a+b+c+d)/2:a1:=(s-a)*(s-b)*(s-c)*(s-d):a2:=sqrt(a*b*c*d):r1:=a2/(a+c):r2:=a2/(b+d):rr:= sqrt((a*b+c*d) * (a*c+b*d) * (a*d+b*c))/(4*a2):if a1>0 and floor(sqrt(a1))=sqrt(a1) and a2 =floor(a2) and a2=sqrt(a1) and r1=floor(r1) and r2=floor(r2) and r1=r2 and rr =floor(rr) then for j from 1 to k do: if sqrt(a2/T[j])=floor(sqrt(a2/T[j]) ) then kk:=1:else fi:od:if kk=0 then k:=k+1:T[k]:=a2: printf ( "%d %d %d %d %d %d %d\n",a2,a,b,c,d,r1,rr):else fi:kk:=0:fi:od:od:od: