A195434 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).

6, 4, 8, 7, 8, 2, 1, 3, 4, 1, 2, 6, 1, 6, 1, 2, 8, 5, 3, 8, 8, 8, 0, 3, 0, 3, 8, 0, 7, 6, 6, 9, 3, 5, 6, 0, 6, 1, 9, 4, 0, 3, 5, 5, 7, 0, 5, 8, 6, 7, 9, 5, 2, 3, 3, 9, 6, 4, 1, 2, 8, 3, 6, 3, 6, 8, 3, 3, 2, 9, 8, 5, 3, 3, 9, 6, 2, 2, 6, 7, 3, 0, 3, 5, 9, 1, 4, 7, 7, 3, 5, 6, 8, 8, 4, 0, 8, 0, 4, 7

Offset: 0

Author

Clark Kimberling, Sep 18 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=0.648782134126161285388803038076693560619403...
		

Links

• Index entries for algebraic numbers, degree 6.

Crossrefs

Cf. A195304, A195435, A195444, A195445.

Programs

Mathematica

a = 1; b = 2; h = 2 a/3; k = b/3;
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195434 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (B) A195435 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (C) A195444 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)

PARI

sqrt(subst((45*t^4 - 126*t^3 + 125*t^2 - 52*t + 8)/(9*t^2 - 12*t + 4), t, polrootsreal(270*x^4 - 738*x^3 + 756*x^2 - 344*x + 56)[1])) \\ Charles R Greathouse IV, Feb 11 2025

PARI

polrootsreal(3645*x^6 + 11421*x^4 + 6219*x^2 - 4913)[2] \\ Charles R Greathouse IV, Feb 11 2025