A197033 Decimal expansion of the shortest distance from the x axis through (2,1) to the line y=x.
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, 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, 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
Offset: 1
Examples
length of Philo line: 1.8442716817001718647799577442735702984134... endpoint on x axis: (2.35321..., 0); see A197032 endpoint on line y=x: (1.73898, 1.73898)
Programs
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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]
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