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A273691 Integer area of primitive cyclic quadrilaterals with integer sides and rational radius. 4
12, 60, 108, 120, 120, 168, 192, 192, 234, 240, 300, 360, 360, 420, 420, 420, 420, 420, 420, 432, 540, 540, 588, 600, 660, 660, 714, 768, 840, 924, 960, 960, 966, 1008, 1008, 1008, 1080, 1080, 1080, 1092, 1134, 1200 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Given 4 segments a,b,c,d, there is a unique circumcircle such that these segments can be placed inside to form cyclic quadrilaterals. There are 3 ways to place these segments: abcd,acbd,adbc.
Primitive means a,b,c,d share no common factor.
The area S = sqrt[(s-a)(s-b)(s-c)(s-d)] where s=(a+b+c+d)/2 is the semiperimeter.
The circumradius R=Sqrt[a b+c d]*Sqrt[a c+b d]*Sqrt[a d+b c]/(4S)
The length of the diagonal separating a-b and c-d is (4S R)/(a b+c d), the other diagonal can be obtain by swapping b,c or swapping b,d.
It follows that if the sides and area are integers, then (any diagonal is rational) <=> (circumradius is rational) <=> (all diagonals are rational).
From empirical observation, the area seems to be a multiple of 6. (If so, the program could be modified to run 6 times as fast.)
LINKS
EXAMPLE
a, b, c, d, S, r
4, 4, 3, 3, 12, 5/2
12, 12, 5, 5, 60, 13/2
14, 13, 13, 4, 108, 65/8
15, 15, 8, 8, 120, 17/2
21, 10, 10, 9, 120, 85/8
24, 24, 7, 7, 168, 25/2
21, 13, 13, 11, 192, 65/6
25, 15, 15, 7, 192, 25/2
24, 20, 15, 7, 234, 25/2
MATHEMATICA
SMax=1200;
Do[
x=S^2/(u v w);
If[u+v+w+x//OddQ, Continue[]];
If[v+w+x<=u, Continue[]];
{a, b, c, d}=(u+v+w+x)/2-{x, w, v, u};
If[GCD[a, b, c, d]>1, Continue[]];
R=(Sqrt[v w+u x]Sqrt[u w+v x]Sqrt[u v+w x])/(4S);
If[R\[NotElement]Rationals, Continue[]];
S(*{a, b, c, d, "", S, R, "", (4S R)/(a d+b c), (4S R)/(a c+b d), (4S R)/(a b+c d)}*)//Sow;
, {S, 1(*6*), SMax, 1(*6*)}(*assuming S mod 6 = 0, set to 6 to run faster*)
, {u, S^2//Divisors//Select[#, S<=#^2&&#<=S&]&}
, {v, S^2/u//Divisors//Select[#, S^2<=u#^3&&u/3<#<=u&]&}
, {w, S^2/(u v)//Divisors//Select[#, S^2<=u v#^2&&(u-v)/2<#<=v&]&}
]//Reap//Last//Last(*//TableForm*)
{S, R, x, a, b, c, d}=.;
CROSSREFS
Sequence in context: A012406 A097191 A338159 * A094807 A120644 A099829
KEYWORD
nonn
AUTHOR
Albert Lau, May 28 2016
STATUS
approved

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Last modified March 28 13:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)