cp's OEIS Frontend

A197136 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x.

%I A197136 #13 Feb 16 2025 02:29:14
%S A197136 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 A197136 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 A197136 7,1,1,6,7,0,4,9,2,9,6,4,8,5,6
%N A197136 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x.
%C A197136 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 A197136 length of Philo line:   3.350162315943772... (see A197137)
%e A197136 endpoint on x axis:   (4.55402, 0)
%e A197136 endpoint on line y=x: (2.93048, 2.93048)
%t A197136 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197136 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197136 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 A197136 m = 1; h = 4; k = 1;(* slope m; point (h,k) *)
%t A197136 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197136 RealDigits[t]  (* A197136 *)
%t A197136 {N[t], 0} (* endpoint on x axis *)
%t A197136 {N[k*t/(k + m*t - m*h)],
%t A197136 N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=mx *)
%t A197136 d = N[Sqrt[f[t]], 100]
%t A197136 RealDigits[d]  (* A197137 *)
%t A197136 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 5}],
%t A197136 ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 5}, {y, 0, 3}],
%t A197136 PlotRange -> {0, 3}, AspectRatio -> Automatic]
%Y A197136 Cf. A197032, A197137, A197008, A195284.
%K A197136 nonn,cons
%O A197136 1,1
%A A197136 _Clark Kimberling_, Oct 10 2011
%E A197136 Incorrect trailing digits removed. - _R. J. Mathar_, Nov 08 2022