%I #5 Mar 30 2012 18:57:52
%S 1,8,4,4,2,7,1,6,8,1,7,0,0,1,7,1,8,6,4,7,7,9,9,5,7,7,4,4,2,7,3,5,7,0,
%T 2,9,8,4,1,3,4,8,7,6,3,3,8,7,7,0,9,5,0,9,1,5,7,4,7,9,4,0,1,7,8,6,4,8,
%U 7,6,8,3,4,3,8,5,3,8,8,6,1,2,4,8,5,0,6,4,4,7,0,9,9,7,5,8,1,8,5,0
%N Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.
%C For discussions and guides to related sequences, see A197032, A197008 and A195284.
%e length of Philo line: 1.8442716817001718647799577442735702984134...
%e endpoint on x axis: (2.35321..., 0); see A197032
%e endpoint on line y=x: (1.73898, 1.73898)
%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 (* root of p[t] minimizes f *)
%t m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197032 *)
%t {N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
%t {N[k*t/(k + m*t - m*h)],
%t N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197033 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%Y Cf. A197032, A197008, A195284.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 10 2011