A197138 - OEIS

%I #9 Nov 08 2022 10:03:19

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

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

%U 2,7,4,3,9,4,6,5,1,7,3,0,1

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

%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.

%e length of Philo line:   2.886117...; see A197139

%e endpoint on x axis:   (3.4883, 0)

%e endpoint on line y=x: (2.80376, 2.80376)

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

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

%t RealDigits[t]  (* A197138 *)

%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)]} (* endpoint on line y=x *)

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

%t RealDigits[d]  (* A197139 *)

%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],

%t ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],

%t PlotRange -> {0, 3}, AspectRatio -> Automatic]

%Y Cf. A197032, A197139, A197008, A195284.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Oct 10 2011

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