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A214097
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Decimal expansion of the constant C = minimum distance between the circumcenter and the excenter of an ex-bicentric quadrilateral whose sides are in an arithmetic progression 1 : 1+d : 1+2d : 1+3d.
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0
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8, 1, 0, 4, 1, 0, 2, 8, 0, 6, 9, 6, 7, 5, 8, 2, 1, 7, 3, 8, 8, 9, 1, 8, 7, 6, 6, 8, 3, 3, 0, 1, 5, 8, 0, 1, 4, 3, 4, 0, 7, 4, 7, 1, 9, 7, 6, 4, 1, 5, 8, 8, 8, 9, 1, 1, 4, 0, 3, 3, 9, 8, 9, 8, 4, 5, 6, 1, 5, 3, 5, 0, 4, 9, 9, 6, 5, 2, 0, 8, 7, 0, 8, 4, 4, 8, 3, 8, 9, 3, 6, 8, 4, 6, 5, 6, 2, 5, 2, 3
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OFFSET
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0,1
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COMMENTS
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Any quadrilateral whose side lengths form an arithmetic progression is ex-tangential. Consequently such an ex-tangential quadrilateral can be configured to be cyclic as well, resulting in an ex-bicentric quadrilateral. Fuss's theorem for ex-bicentric quadrilaterals defines the distance between the circumcenter and the excenter. If the ratio of the sides of this quadrilateral is 1 : 1+d : 1+2d : 1+3d, then there is a minimum distance between these two centers at some value for d.
An algebraic number of degree 22 and denominator 54; minimal polynomial 509607936x^22 - 9384230912x^20 - 7214465024x^18 + 11738382336x^16 + 5482373120x^14 - 4279234560x^12 - 369279488x^10 - 45531392x^8 + 2774880x^6 + 158928x^4 + 4068x^2 - 45. - Charles R Greathouse IV, Apr 21 2016
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LINKS
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FORMULA
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Area K=Sqrt[(1+d)(1+2d)(1+3d)], Exradius r=K/|2d|, Circumradius R=Sqrt[((1+d)+(1+2d)(1+3d))*((1+2d)+(1+d)(1+3d))*((1+3d)+(1+d)(1+2d))]/(4K) and the distance between the circumcenter and the excenter is given by Fuss's theorem x=Sqrt[R^2+r^2+r*Sqrt[4R^2+r^2]]. Constant C = minimum value for x.
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EXAMPLE
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The minimum value C=0.8104102806967582... when d=-0.3095759507409512...
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MATHEMATICA
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k=Sqrt[(1+d)(1+2d)(1+3d)]; s=Sqrt[((1+d) + (1+2d)(1+3d)) ((1+2d)+(1+d)(1+3d)) ((1+3d)+(1+d)(1+2d))]/(4k); r=k/Abs[2d]; N[Minimize[{Sqrt[s^2+r^2+r*Sqrt[4s^2+r^2]], -1/3<d<0}, {d}], 100]
c = Sqrt[Root[Function[d, 243*d^11 - 17899*d^10 - 55042*d^9 + 358227*d^8 + 669235*d^7 - 2089470*d^6 - 721249*d^5 - 355714*d^4 + 86715*d^3 + 19866*d^2 + 2034*d - 90], 3]]/2; RealDigits[c, 10, 100][[1]] (* Jean-François Alcover, Feb 20 2014, using the derivative of the distance *)
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PROG
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(PARI) polrootsreal(509607936*x^22 - 9384230912*x^20 - 7214465024*x^18 + 11738382336*x^16 + 5482373120*x^14 - 4279234560*x^12 - 369279488*x^10 - 45531392*x^8 + 2774880*x^6 + 158928*x^4 + 4068*x^2 - 45)[8] \\ Charles R Greathouse IV, Apr 21 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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