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%I #7 Mar 30 2012 18:57:52
%S 4,9,2,5,4,5,5,2,2,0,4,3,5,0,0,2,8,4,4,0,9,7,4,9,8,0,2,1,9,0,5,0,2,7,
%T 3,0,7,8,7,4,1,2,9,9,2,1,1,4,9,9,6,8,6,7,1,5,9,4,8,6,0,3,3,2,4,7,8,5,
%U 2,6,5,8,1,6,3,1,3,8,3,0,0,5,0,8,7,6,8,0,1,9,9,1,3,3,0,7,6,7,6,2
%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) 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: 4.70800001749646..; see A197147
%e endpoint on x axis: (4.92546, 0)
%e endpoint on line y=2x: (1.72768, 3.45536)
%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 = 4; k = 1; (* slope m, point (h,k) *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197146 *)
%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] (* A197147 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .004, {x, 0, 5}, {y, 0, 3}], PlotRange -> {0, 4}, AspectRatio -> Automatic]
%Y Cf. A197032, A197147, A197008, A195284.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 11 2011
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