A195487 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)=(sqrt(7),3,4).

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

Offset: 1

Author

Clark Kimberling, Sep 19 2011

Keywords

• nonn
• cons

Comments

See A195304 for definitions and a general discussion.

Examples

			(A)=1.648040734595518812330740700...
		

Crossrefs

Cf. A195304, A195488, A195489, A195490.

Programs

Mathematica

a = Sqrt[7]; b = 3; 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) A195487 *)
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) A195488 *)
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) A195489  *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)