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%I A197139 #19 May 03 2024 15:49:44
%S A197139 2,8,8,6,1,1,7,1,0,5,8,9,8,0,0,1,2,9,1,5,3,6,7,2,6,5,3,2,0,0,9,5,1,1,
%T A197139 4,1,4,5,1,7,1,7,7,6,1,7,4,7,7,3,9,4,8,5,3,3,8,8,0,7,7,5,4,2,9,4,9,9,
%U A197139 1,5,0,7,4,1,3,0,8,4,2,4,6,6,2,4,9,4,9,2,7,6,4,3,9,9,0,1,8,3,2
%N A197139 Decimal expansion of the shortest distance from the x axis through (3,2) to the line y = x.
%C A197139 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 A197139 Any Philo line can be constructed using the intersections of circles and hyperbolas. In this case, the Philo line passes though the two points at which the circle (x - 3/2)^2 + (y - 1)^2 = 13/4 and the hyperbola x*y - y^2 = 2 intersect. The circle has segment OP as diameter, where O(0,0) is the origin and P is the point (3,2). The asymptotes of the hyperbola are the x axis and the line y = x. Point P is one of the two points at which the circle and the hyperbola intersect. - _Raul Prisacariu_, Apr 06 2024
%e A197139 Length of Philo line:   2.8861171058980012915367...
%e A197139 Endpoint on x axis:     (3.4883, 0); see A197138
%e A197139 Endpoint on line y = x: (2.80376, 2.80376)
%t A197139 f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t A197139 g[t_] := D[f[t], t]; Factor[g[t]]
%t A197139 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 A197139 m = 1; h = 3; k = 2;(* slope m; point (h,k) *)
%t A197139 t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t A197139 RealDigits[t]  (* A197138 *)
%t A197139 {N[t], 0} (* endpoint on x axis *)
%t A197139 {N[k*t/(k + m*t - m*h)],
%t A197139 N[m*k*t/(k + m*t - m*h)]} (* endpoint on line y=x *)
%t A197139 d = N[Sqrt[f[t]], 100]
%t A197139 RealDigits[d]  (* this sequence *)
%t A197139 Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],
%t A197139 ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 4}, {y, 0, 3}],
%t A197139 PlotRange -> {0, 3}, AspectRatio -> Automatic]
%Y A197139 Cf. A197032, A197138, A197008, A195284.
%K A197139 nonn,cons
%O A197139 1,1
%A A197139 _Clark Kimberling_, Oct 10 2011
%E A197139 Last digit removed (repr. truncated, not rounded up) by _R. J. Mathar_, Nov 08 2022

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