A195491 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,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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

Offset: 0

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=0.62958106138771604404549587568854069...
		

Crossrefs

Cf. A195304, A195492, A195493, A195494.

Programs

Mathematica

a = 1; b = Sqrt[GoldenRatio]; 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) A195491 *)
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) A195492 *)
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, 4]
RealDigits[%, 10, 100] (* (C) A195493 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)