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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

Table of n, a(n) for n=1..42.

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: A012657 A012406 A097191 * A094807 A120644 A099829

Adjacent sequences:  A273688 A273689 A273690 * A273692 A273693 A273694

KEYWORD

nonn

AUTHOR

Albert Lau, May 28 2016

STATUS

approved

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Last modified April 6 14:08 EDT 2020. Contains 333276 sequences. (Running on oeis4.)