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

3

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