%I #8 Nov 08 2022 11:44:34

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

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

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

%N Decimal expansion of the shortest distance from the x axis through (2,1) 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: 3.160946973065...

%e endpoint on x axis: (2.85106, 0); see A197150

%e endpoint on line y=3x: (0.802397, 2.40719)

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

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

%t RealDigits[t] (* A197150 *)

%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] (* A197151 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],

%t PlotRange -> {0, 2.5}, AspectRatio -> Automatic]

%Y Cf. A197032, A197150, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2011