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%I #23 Aug 14 2023 10:33:25
%S 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,
%T 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,
%U 1,8,9,7,2,3,2,2,0,9,2,0,8,5,9,1,6,5,8,0,3
%N Decimal expansion of the x-intercept of the shortest segment from the positive x axis through (2,1) to the line y=x.
%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.
%C Philo lines from positive x axis through (h,k) to line y=mx:
%C m......h......k....x-intercept.....distance
%C 1......2......1.......A197032......A197033
%C 1......3......1.......A197034......A197035
%C 1......4......1.......A197136......A197137
%C 1......3......2.......A197138......A197139
%C 2......1......1.......A197140......A197141
%C 2......2......1.......A197142......A197143
%C 2......3......1.......A197144......A197145
%C 2......4......1.......A197146......A197147
%C 3......1......1.......A197148......A197149
%C 3......2......1.......A197150......A197151
%C 1/2....3......1.......A197152......A197153
%C 1/2....4......1.......A197154......A197155
%H R. J. Mathar, <a href="/A197032/a197032.pdf">OEIS A197032</a>, Nov. 8, 2022
%H M. F. Hasler, <a href="https://docs.google.com/drawings/d/1MjqKB9QvbvzibYBmPozumEDDvwzOSbXwI1GClR3cvv4/edit?usp=sharing">Philo line - oeis.org/A197032</a> (google drawing), Nov. 8, 2022
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Philo_line">Philo line</a>
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F x = 2 + tan phi where 1 + 2 tan phi = 1/(sin phi + cos phi), whence x = 1 + A357469 = the only real root of x^3 - 4*x^2 + 6*x - 5. - _M. F. Hasler_, Nov 08 2022
%e length of Philo line: 1.8442716817001... (see A197033)
%e endpoint on x axis: (2.35321..., 0)
%e endpoint on y=x: (1.73898, 1.73898)
%p Digits := 140 ;
%p x^3-4*x^2+6*x-5 ;
%p fsolve(%=0) ; # _R. J. Mathar_, Nov 08 2022
%t f[t_] := (t - k*t/(k + m*t - m*h))^2 + (m*k*t/(k + m*t - m*h))^2;
%t g[t_] := D[f[t], t]; Factor[g[t]]
%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 *)
%t m = 1; h = 2; k = 1; (* m=slope; (h,k)=point *)
%t t = t1 /. FindRoot[p[t1] == 0, {t1, 1, 2}, WorkingPrecision -> 100]
%t RealDigits[t] (* A197032 *)
%t {N[t], 0} (* lower endpoint of minimal segment [Philo line] *)
%t {N[k*t/(k + m*t - m*h)],
%t N[m*k*t/(k + m*t - m*h)]} (* upper endpoint *)
%t d = N[Sqrt[f[t]], 100]
%t RealDigits[d] (* A197033 *)
%t Show[Plot[{k*(x - t)/(h - t), m*x}, {x, 0, 2.5}],
%t ContourPlot[(x - h)^2 + (y - k)^2 == .003, {x, 0, 3}, {y, 0, 3}], PlotRange -> {0, 2}, AspectRatio -> Automatic]
%o (PARI) solve(x=2,3, x^3 - 4*x^2 + 6*x - 5)
%Y Cf. A197033, A197008, A195284.
%Y Cf. A357469 (= this constant - 1).
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Oct 10 2011
%E Invalid trailing digits corrected by _R. J. Mathar_, Nov 08 2022
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