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Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.
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%I #7 Nov 08 2022 12:15:49

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

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

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

%N Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.

%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 x^3/2 -5*x^2/2 +9*x/2 -5. - _R. J. Mathar_, Nov 08 2022

%e length of Philo line: 1.481506505...; see A197153

%e endpoint on x axis: (3.15091, 0)

%e endpoint on line y=3x: (2.92984, 1.46492)

%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 = 1/2; h = 3; k = 1;(* slope m, point (h,k) *)

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

%t RealDigits[t] (* A197152 *)

%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=x/2 *)

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

%t RealDigits[d] (* A197153 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3.5}],

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

%t PlotRange -> {0, 1.5}, AspectRatio -> Automatic]

%Y Cf. A197032, A197153, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 11 2011

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