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%I #6 Mar 30 2012 18:57:52
%S 3,8,2,8,9,1,1,1,4,1,5,4,2,9,4,3,6,5,3,2,1,9,8,8,2,2,4,1,3,9,6,4,8,6,
%T 7,2,1,7,2,4,4,5,0,5,3,9,0,2,8,4,8,7,2,0,0,6,8,2,2,8,6,6,4,6,4,8,7,9,
%U 4,9,4,6,6,2,6,1,3,2,4,9,7,5,7,1,7,5,9,4,6,9,1,5,9,2,6,0,8,4,5,7
%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=2x.
%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: 3.7423891424451...; see A197145
%e endpoint on x axis: (3.82891, 0)
%e endpoint on line y=2x: (1.44062, 2.88124)
%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 = 2; h = 3; k = 1;(* slope m, point (h,k) *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197144 *)
%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=2x *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197145 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]
%Y Cf. A197032, A197145, A197008, A195284.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 11 2011
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