%I #6 Mar 30 2012 18:57:52

%S 2,6,9,1,4,1,3,9,7,9,7,8,1,7,3,6,7,9,1,9,2,8,8,6,5,6,0,5,7,5,3,2,2,5,

%T 2,1,8,3,8,5,7,6,7,6,4,6,9,2,4,6,8,9,7,0,9,7,1,2,4,7,6,5,3,6,6,0,0,4,

%U 2,2,1,8,8,2,5,9,8,6,2,1,0,6,1,9,1,0,1,9,6,9,9,3,8,3,3,7,6,0,0,4

%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) 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: 2.7463941076100...; see A197143

%e endpoint on x axis: (2.69141, 0)

%e endpoint on line y=2x: (1.1295, 2.25901)

%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 = 2; k = 1;(* slope m, point (h,k) *)

%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]

%t RealDigits[t] (* A197142 *)

%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] (* A197143 *)

%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, 2.5}, AspectRatio -> Automatic]

%Y Cf. A197032, A197143, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2011