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 A197032 Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x. 24
 2, 3, 5, 3, 2, 0, 9, 9, 6, 4, 1, 9, 9, 3, 2, 4, 4, 2, 9, 4, 8, 3, 1, 0, 1, 3, 3, 2, 5, 7, 7, 3, 8, 8, 4, 5, 7, 2, 7, 0, 7, 0, 5, 6, 1, 3, 8, 5, 6, 8, 4, 6, 8, 2, 6, 8, 0, 6, 6, 9, 3, 0, 4, 2, 6, 5, 1, 5, 1, 8, 9, 7, 2, 3, 2, 2, 0, 9, 2, 0, 8, 5, 9, 1, 8, 1, 6, 4, 7, 0, 6, 9, 1, 1, 1, 6, 4, 5, 4, 3 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. Philo lines from positive x axis through (h,k) to line y=mx: m......h......k....x-intercept.....distance 1......2......1.......A197032......A197033 1......3......1.......A197034......A197035 1......4......1.......A197036......A197037 1......3......2.......A197038......A197039 2......1......1.......A197040......A197041 2......2......1.......A197042......A197043 2......3......1.......A197044......A197045 2......4......1.......A197046......A197047 3......1......1.......A197048......A197049 3......2......1.......A197050......A197051 1/2....3......1.......A197052......A197053 1/2....4......1.......A197054......A197055 LINKS EXAMPLE length of Philo line:  1.8442716817001... (see A197033) endpoint on x axis: (2.35321..., 0) endpoint on y=x:    (1.73898, 1.73898) MATHEMATICA f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2; g[t_] := D[f[t], t]; Factor[g[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 *) m = 1; h = 2; k = 1; (* m=slope; (h, k)=point *) t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100] RealDigits[t]  (* A197032 *) {N[t], 0} (* lower endpoint of minimal segment [Philo line] *) {N[k*t/(k + m*t - m*h)], N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *) d = N[Sqrt[f[t]], 100] RealDigits[d] (* A197033 *) Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}], ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic] CROSSREFS Cf. A197033, A197008, A195284. Sequence in context: A238684 A202694 A123221 * A321781 A254862 A322235 Adjacent sequences:  A197029 A197030 A197031 * A197033 A197034 A197035 KEYWORD nonn,cons AUTHOR Clark Kimberling, Oct 10 2011 STATUS approved

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Last modified January 25 03:49 EST 2020. Contains 331241 sequences. (Running on oeis4.)