OFFSET
0,1
COMMENTS
See A195284 for definitions and a general discussion. This constant is the maximum of Philo(ABC,I) over all triangles ABC.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
FORMULA
Equals (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))).
EXAMPLE
Philo(ABC,I)=0.614058971032212611546384892539385408260...
MATHEMATICA
a = 1; b = 1; c = Sqrt[2];
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, 1]
RealDigits[%, 10, 100] (* (A) A195301 *)
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] (* (B)=(A) *)
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, 1]
RealDigits[%, 10, 100] (* (C) A163960 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, I), A195303 *)
PROG
(PARI) (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))) \\ Michel Marcus, Jul 27 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 14 2011
STATUS
approved