cp's OEIS Frontend

A197154 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x/2.

%I A197154 #11 Feb 16 2025 02:29:20
%S A197154 4,2,2,3,6,0,3,3,0,4,1,9,3,9,8,1,1,4,7,6,0,7,9,3,9,5,2,5,9,6,1,6,9,0,
%T A197154 1,4,2,0,9,3,1,8,6,8,4,1,9,3,2,0,8,6,3,1,1,7,6,6,3,2,5,7,8,0,7,1,2,4,
%U A197154 8,7,6,2,3,0,6,5,6,6,0,5,7,7,8,3,5,2,3,0,8,5,0,5,5,3,0,6,3,0,8
%N A197154 Decimal expansion of the x-intercept of the shortest segment from the x axis through (4,1) to the line y=x/2.
%C A197154 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.
%e A197154 length of Philo line:    1.94634640...; see A197155
%e A197154 endpoint on x axis:    (4.2236, 0)
%e A197154 endpoint on line y=3x: (3.79888, 1.89944)
%t A197154 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197154 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197154 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 A197154 m = 1/2; h = 4; k = 1;(* slop m, point (h,k) *)
%t A197154 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197154 RealDigits[t]  (* A197154 *)
%t A197154 {N[t], 0} (* endpoint on x axis *)
%t A197154 {N[k*t/(k + m*t - m*h)],
%t A197154 N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=x/2 *)
%t A197154 d = N[Sqrt[f[t]], 100]
%t A197154 RealDigits[d]  (* A197155 *)
%t A197154 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4.5}],
%t A197154 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4.5}, {y, 0, 3}],
%t A197154 PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y A197154 Cf. A197032, A197155, A197008, A195284.
%K A197154 nonn,cons
%O A197154 1,1
%A A197154 _Clark Kimberling_, Oct 11 2011
%E A197154 Last digit removed (repr. truncated, not rounded up) by _Georg Fischer_, Nov 11 2022