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Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x.
3

%I #9 Nov 08 2022 09:21:40

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

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

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

%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x.

%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 A197008 and A195284.

%e length of Philo line: 3.350162315943772... (see A197037)

%e endpoint on x axis: (4.55402, 0)

%e endpoint on line y=x: (2.93048, 2.93048)

%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 = 1; h = 4; k = 1;(* slope m; point (h,k) *)

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

%t RealDigits[t] (* A197136 *)

%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)]} (* endpoint on line y=mx *)

%t d = N[Sqrt[f[t]], 100]

%t RealDigits[d] (* A197137 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 5}, {y, 0, 3}],

%t PlotRange -> {0, 3}, AspectRatio -> Automatic]

%Y Cf. A197032, A197137, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 10 2011

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