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.

A root of the polynomial x^3-7*x^2+18*x-20. - R. J. Mathar, Nov 08 2022

EXAMPLE

length of Philo line: 2.60819402496101...; see A197035

endpoint on x axis: (3.47797, 0)

endpoint on line y=x: (2.35321, 2.35321)

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 = 1; h = 3; k = 1; (* slope m; point (h, k) *)

t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]

RealDigits[t] (* A197034 *)

{N[t], 0} (* endpoint on x axis *)

{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] (* A197035 *)

Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 4}],

ContourPlot[(x - h)^2 + (y - k)^2 == .002, {x, 0, 3.5}, {y, 0, 3}], PlotRange -> {0, 3}, AspectRatio -> Automatic]

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Oct 10 2011

EXTENSIONS

Last digit removed (representation truncated, not rounded up). - R. J. Mathar, Nov 08 2022

STATUS

approved