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%I #6 Mar 30 2012 18:57:52
%S 1,6,0,4,7,9,3,6,1,8,4,6,2,1,3,9,9,0,7,3,7,8,3,1,7,9,5,0,7,1,7,9,6,1,
%T 8,4,6,7,1,5,4,4,9,2,1,9,9,9,1,2,8,6,0,7,7,8,6,3,6,2,9,2,2,1,4,9,2,1,
%U 6,3,7,2,6,1,9,1,2,6,0,4,2,1,6,6,7,9,9,7,0,2,2,8,4,7,0,1,4,7,7,2
%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) to the line y=3x.
%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: 1.999158399580...; see A197149
%e endpoint on x axis: (1.60479, 0)
%e endpoint on line y=3x: (0.570212, 1.71064)
%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 = 3; h = 1; k = 1;(* slope m, point (h,k) *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197148 *)
%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=3x *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197149 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y Cf. A197032, A197149, A197008, A195284.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 11 2011
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