cp's OEIS Frontend

A197152 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.

%I A197152 #7 Nov 08 2022 12:15:49
%S A197152 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 A197152 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 A197152 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 A197152 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x/2.
%C A197152 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 A197152 A root of the polynomial x^3/2 -5*x^2/2 +9*x/2 -5. - _R. J. Mathar_, Nov 08 2022
%e A197152 length of Philo line:    1.481506505...; see A197153
%e A197152 endpoint on x axis:    (3.15091, 0)
%e A197152 endpoint on line y=3x: (2.92984, 1.46492)
%t A197152 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197152 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197152 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 A197152 m = 1/2; h = 3; k = 1;(* slope m, point (h,k) *)
%t A197152 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197152 RealDigits[t] (* A197152 *)
%t A197152 {N[t], 0} (* endpoint on x axis *)
%t A197152 {N[k*t/(k + m*t - m*h)],
%t A197152 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=x/2 *)
%t A197152 d = N[Sqrt[f[t]], 100]
%t A197152 RealDigits[d]  (* A197153 *)
%t A197152 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3.5}],
%t A197152 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
%t A197152 PlotRange -> {0, 1.5}, AspectRatio -> Automatic]
%Y A197152 Cf. A197032, A197153, A197008, A195284.
%K A197152 nonn,cons
%O A197152 1,1
%A A197152 _Clark Kimberling_, Oct 11 2011
%E A197152 Incorrect trailing digits deleted. - _R. J. Mathar_, Nov 08 2022