OFFSET
1,2
COMMENTS
The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P. Philo lines are not generally Euclidean-constructible.
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Suppose that P lies inside a triangle ABC. Let (A) denote the shortest length of segment from AB through P to AC, and likewise for (B) and (C). The Philo sum for ABC and P is here introduced as s=(A)+(B)+(C), and the Philo number for ABC and P, as s/(a+b+c), denoted by Philo(ABC,P).
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Listed below are examples for which P=G (the centroid); in this list, r'n means sqrt(n) and t=(1+sqrt(5))/2 (the golden ratio).
a....b...c........(A).......(B)........(C)...Philo(ABC,G)
A similar list for P=incenter is given at A195284.
EXAMPLE
(A)=1.89630056630920201475386720365481991708010328...
MATHEMATICA
a = 3; b = 4; 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) A195304 *)
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) A195305 *)
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) A195306 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC, G) A195411 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 18 2011
STATUS
approved