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A341859 Decimal expansion of 4 - (8/5)*sqrt(5).

%I A341859 #35 Mar 09 2021 08:16:26
%S A341859 4,2,2,2,9,1,2,3,6,0,0,0,3,3,6,4,8,5,7,4,5,3,2,2,1,3,0,0,2,9,9,5,8,0,
%T A341859 2,3,2,9,5,0,1,0,6,2,4,6,2,1,5,5,8,8,4,1,1,6,6,5,6,4,4,0,7,3,4,3,1,6,
%U A341859 6,5,1,8,9,7,9,5,1,2,1,6,0,9,3,6,9,3,6,9,4,6,5,9,3,9,4,8,3,6
%N A341859 Decimal expansion of 4 - (8/5)*sqrt(5).
%C A341859 In a triangle inscribed in a unit circle this is the maximal value of its inradius, such that a minimal closed Steiner chain of circles (10 circles) can be sandwiched between the incircle and circumcircle of the triangle.
%C A341859 It can be found as follows.
%C A341859 The squared distance between the centers of the two chain-defining circles is known to be d^2 = (R-r)^2 - 4*r*R*tan(Pi/n)^2.
%C A341859 On the other hand, the squared distance between the circumcenter and the incenter of triangle is known to be d^2 = R*(R-2*r).
%C A341859 Thus, in order to make a valid closed chain of circles, the inradius of triangle inscribed in the unit circle must be equal to 4*tan(Pi/n)^2.
%C A341859 Given that the maximum of such inradius is 0.5, the minimal number of chained circles is n=10, which gives the maximal value r = 4*tan(Pi/10)^2 = 0.42... < 0.5.
%D A341859 Liang-Shin Hahn. Complex Numbers and Geometry (Mathematical Association of America Textbooks). The Mathematical Association of America, 1994, 140-141.
%H A341859 Wikipedia, <a href="https://en.wikipedia.org/wiki/Steiner_chain">Steiner Chain</a>.
%F A341859 Equals 4*A019916^2 = 4*tan(Pi/10)^2 = 4 - (8/5)*sqrt(5) = (4/5)*(7 - 4*phi) = (4/5)*(7 - 4*A001622), where phi is the golden ratio from A001622.
%e A341859 0.4222912360003364857453221300299580232950106246215588411665644073...
%t A341859 RealDigits[4*Tan[18 Degree]^2, 10, 120][[1]]
%o A341859 (PARI) 4-8/5*sqrt(5)
%Y A341859 Cf. A019916, A001622.
%K A341859 nonn,cons
%O A341859 0,1
%A A341859 _Gleb Koloskov_, Mar 07 2021