A195493 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

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

Offset: 0

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			(C)=0.759310778373734956811842690497767...
		

Crossrefs

Cf. A195304.

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 *)