A197144 - OEIS

%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