editing

approved

Integer area of primitive cylic quadrilateral cyclic quadrilaterals with integer side sides and rational radius.

Primitive means a,b,c,d share no common factorsfactor.

The area S = sqrt[(s-a)(s-b)(s-c)(s-d)] where s=(a+b+c+d)/2 is the semiperimeter.

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 side 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 can could be modified to run 6 times fasteras fast.)

proposed

editing

editing

proposed

allocated for Albert LauInteger area of primitive cylic quadrilateral with integer side and rational radius.

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

1,1

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 factors.

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 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 side and area are integers, then (any diagonal is rational) <=> (circumradius is rational) <=> (all diagonals are rational).

From empirical observation, the area seems to be multiple of 6. (If so, the program can be modified to run 6 times faster)

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

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}=.;

allocated

nonn

Albert Lau, May 28 2016

approved

editing

allocated for Albert Lau

allocated

approved