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%I #5 Mar 30 2012 18:57:52
%S 1,9,4,6,3,4,6,4,0,2,3,7,8,4,8,3,8,5,6,1,6,6,4,0,9,1,1,4,2,3,0,0,8,0,
%T 6,8,1,8,5,8,2,1,0,6,7,1,1,7,6,0,3,8,5,7,0,1,8,9,2,3,8,5,0,9,1,0,4,9,
%U 9,8,9,5,6,0,1,8,8,6,8,0,1,9,1,0,7,7,4,4,3,2,0,7,0,6,5,2,2,4,1,4
%N Decimal expansion of the shortest distance from the x axis through (4,1) to the line y=x/2.
%C The shortest segment from one side of an angle T through a point P inside T is called the Philo line of P in T. For discussions and guides to related sequences, see A197032, A197008 and A195284.
%e length of Philo line: 1.94634640...
%e endpoint on x axis: (4.2236, 0); see A197154
%e endpoint on line y=3x: (3.79888, 1.89944)
%t f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t g[t_] := D[f[t], t]; Factor[g[t]]
%t p[t_] := h^2 k + k^3 - h^3 m - h k^2 m - 3 h k t + 3 h^2 m t + 2 k t^2 - 3 h m t^2 + m t^3
%t m = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197154 *)
%t {N[t], 0} (* endpoint on x axis] *)
%t {N[k*t/(k + m*t - m*h)],
%t N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=x/2 *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197155 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
%t PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y Cf. A197032, A197155, A197008, A195284.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 11 2011
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