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A197150
Decimal expansion of the x-intercept of the shortest segment from the x axis through (2,1) to the line y=3x.
3
2, 8, 5, 1, 0, 5, 9, 7, 9, 0, 6, 2, 9, 8, 6, 2, 6, 3, 9, 8, 2, 6, 0, 7, 2, 3, 4, 0, 8, 5, 3, 6, 2, 6, 8, 9, 3, 3, 3, 9, 0, 9, 3, 5, 8, 1, 7, 2, 2, 1, 0, 1, 7, 3, 2, 2, 7, 0, 3, 3, 5, 9, 2, 6, 3, 9, 6, 4, 6, 0, 5, 4, 3, 6, 4, 2, 1, 0, 3, 0, 0, 7, 6, 0, 4, 2
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 A197032, A197008 and A195284.
A root of the polynomial 3*x^3-16*x^2+30*x-25. - R. J. Mathar, Nov 08 2022
EXAMPLE
length of Philo line: 3.160946973065...; see A197151
endpoint on x axis: (2.85106, 0)
endpoint on line y=3x: (0.802397, 2.40719)
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
m = 3; h = 2; k = 1; (* slope m, point (h, k) *)
t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
RealDigits[t] (* A197150 *)
{N[t], 0} (* endpoint on x axis *)
{N[k*t/(k + m*t - m*h)],
N[m*k*t/(k + m*t - m*h)]} (* endpt on line y=3x *)
d = N[Sqrt[f[t]], 100]
RealDigits[d] (* A197151 *)
Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 3}],
ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 4}, {y, 0, 3}],
PlotRange -> {0, 2.5}, AspectRatio -> Automatic]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 11 2011
EXTENSIONS
Incorrect trailing digits removed. - R. J. Mathar, Nov 08 2022
STATUS
approved