%I #9 Nov 08 2022 12:14:56

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

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

%U 6,3,7,3,8,9,3,2,7,7,8,9,2,9,4,2

%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (1,1) 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.

%C A root of the polynomial 2*x^3-4*x^2+3*x-2. - _R. J. Mathar_, Nov 08 2022

%e length of Philo line: 1.6736473041529...; see A197139

%e endpoint on x axis: (1.44062, 0)

%e endpoint on line y=2x: (0.765782, 1.53156)

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

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

%t RealDigits[t] (* A197140 *)

%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] (* A197141 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 3}], PlotRange -> {0, 1.7}, AspectRatio -> Automatic]

%Y Cf. A197032, A197141, A197008, A195284.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 11 2011

%E Incorrect trailing digits removed. - _R. J. Mathar_, Nov 08 2022