A273771 Integer area of primitive tangential quadrilateral with integer sides, integer inradius and rational diagonals.

168, 408, 420, 528, 588, 600, 660, 1008

Offset

1,1

Comments

A tangential quadrilateral is a quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius.

The area S of a tangential quadrilateral is given by S = r s where s is the semiperimeter and r is the inradius.

The sides of a tangential quadrilateral satisfy s = a + c = b + d where a,c and b,d are opposite sides.

Let D^2 = a b c d - S^2 (D can be positive or negative), then the distance from the tangent point on a(or b) to the vertex point between a,b is given by (ab-D)/s. Similar formula is given for changing a-b to b-c, c-d and d-a.

As a consequences of above formula, a b c d >= S^2.

The diagonal separating ad and bc is p=Sqrt[(a-d)^2+(4S^2)/(a d+b c+2D)]

The diagonal separating ab and cd is q=Sqrt[(a-b)^2+(4S^2)/(a b+c d-2D)]

Links

Table of n, a(n) for n=1..8.

Example

a,   b,   c,  d,  S,    r,  p,   q
15,  15,  13, 13, 168,  6,  14,  24
26,  26,  25, 25, 408,  8,  17,  48
25,  25,  17, 17, 420,  10, 28,  30
26,  26,  22, 22, 528,  11, 40,  132/5
28,  28,  21, 21, 588,  12, 35,  168/5
25,  25,  25, 25, 600,  12, 40,  30
39,  30,  16, 25, 660,  12, 34,  39
102, 102, 10, 10, 1008, 9,  104, 252/13

Mathematica

SMax=500;
Do[
  If[a==c&&\[CapitalDelta]<0, Continue[]];
  If[GCD[a, b, s, r]>1, Continue[]];
  If[b c+\[CapitalDelta]<=0||c d-\[CapitalDelta]<=0, Continue[]];
  If[!{p=Sqrt[(a-d)^2+(4S^2)/(a d+2\[CapitalDelta]+b c)],
       q=Sqrt[(a-b)^2+(4S^2)/(a b-2\[CapitalDelta]+c d)]
      }\[Element]Rationals, Continue[]];
  S(*{a, b, c, d, S, r, \[CapitalDelta], p, q}*)//Sow;
  , {S, SMax}, {s, S//Divisors//Select[#, #^2>=4S&]&}, {r, {S/s}}
  , {a, s/2//Ceiling, s}, {c, {s-a}}
  , {b, s/2//Ceiling, a}, {d, {s-b}}
  , {\[CapitalDelta], Select[{1, -1}Sqrt[a b c d-S^2], IntegerQ]//Union}
]//Reap//Last//Last(*//TableForm*)
{p, q}=.;

Crossrefs

Sequence in context: A230479 A105915 A158219 * A210207 A303082 A247721

Adjacent sequences: A273768 A273769 A273770 * A273772 A273773 A273774

Keyword

nonn

Author

Albert Lau, Jun 03 2016

Status

approved