A195285 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

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

Offset: 0

Author

Clark Kimberling, Sep 14 2011

Keywords

• nonn
• cons

Comments

See A195284 for a definition of Philo(ABC,I) and general discussion.

Examples

			Philo(ABC,I)=0.59772335075207494572...
		

Crossrefs

Cf. A002163, A010466, A195284.

Programs

Mathematica

a = 3; b = 4; c = 5;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
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) A195284 *)
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] (* (B) A002163 *)
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] (* (C) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)