cp's OEIS Frontend

A197034 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x.

%I A197034 #14 Nov 08 2022 13:41:31
%S A197034 3,4,7,7,9,6,7,2,4,3,0,0,9,0,1,2,4,7,4,6,4,6,9,2,5,0,8,1,3,4,2,1,7,5,
%T A197034 1,0,1,4,4,7,5,4,9,5,5,2,7,5,8,1,9,3,4,4,4,2,3,5,9,0,9,9,3,8,6,0,4,6,
%U A197034 0,4,0,6,3,1,9,6,0,1,1,8,7,6,9,8,4,9,7,7,5,3,6,2,6,5,5,3,0,8,5
%N A197034 Decimal expansion of the x-intercept of the shortest segment from the x axis through (3,1) to the line y=x.
%C A197034 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.
%C A197034 A root of the polynomial x^3-7*x^2+18*x-20. - _R. J. Mathar_, Nov 08 2022
%e A197034 length of Philo line:   2.60819402496101...; see A197035
%e A197034 endpoint on x axis:   (3.47797, 0)
%e A197034 endpoint on line y=x: (2.35321, 2.35321)
%t A197034 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197034 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197034 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 A197034 m = 1; h = 3; k = 1;  (* slope m; point (h,k) *)
%t A197034 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197034 RealDigits[t]  (* A197034 *)
%t A197034 {N[t], 0} (* endpoint on x axis *)
%t A197034 {N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
%t A197034 d = N[Sqrt[f[t]], 100]
%t A197034 RealDigits[d] (* A197035 *)
%t A197034 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
%t A197034 ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]
%Y A197034 Cf. A197032, A197035, A197008, A195284.
%K A197034 nonn,cons
%O A197034 1,1
%A A197034 _Clark Kimberling_, Oct 10 2011
%E A197034 Last digit removed (representation truncated, not rounded up). - _R. J. Mathar_, Nov 08 2022