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Decimal expansion of shortest length of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5); i.e., decimal expansion of 2*sqrt(10)/3.
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%I #41 Jul 18 2021 11:15:53

%S 2,1,0,8,1,8,5,1,0,6,7,7,8,9,1,9,5,5,4,6,6,5,9,2,9,0,2,9,6,2,1,8,1,2,

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

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

%N Decimal expansion of shortest length of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5); i.e., decimal expansion of 2*sqrt(10)/3.

%C Apart from the first digit, the same as A176219 (decimal expansion of 2+2*sqrt(10)/3).

%C 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.

%C ...

%C 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 number for ABC and P is here introduced as the normalized sum ((A)+(B)+(C))/(a+b+c), denoted by Philo(ABC,P).

%C ...

%C Listed below are examples for which P=incenter (the center, I, of the circle inscribed in ABC, the intersection of the angle bisectors of ABC); in this list, r'x means sqrt(x), and t=(1+sqrt(5))/2 (the golden ratio).

%C a....b....c.......(A).......(B).......(C)....Philo(ABC,I)

%C 3....4....5.....A195284...A002163...A010466...A195285

%C 5....12...13....A195286...A195288...A010487...A195289

%C 7....24...25....A195290...A010524...15/2......A195292

%C 8....15...17....A195293...A195296...A010524...A195297

%C 28...45...53....A195298...A195299...A010466...A195300

%C 1....1....r'2...A195301...A195301...A163960...A195303

%C 1....2....r'5...A195340...A195341...A195342...A195343

%C 1....3....r'10..A195344...A195345...A195346...A195347

%C 2....3....r'13..A195355...A195356...A195357...A195358

%C 2....5....r'29..A195359...A195360...A195361...A195362

%C r'2..r'3..r'5...A195365...A195366...A195367...A195368

%C 1....r'2..r'3...A195369...A195370...A195371...A195372

%C 1....r'3..2.....A195348...A093821...A120683...A195380

%C 2....r'5..3.....A195381...A195383...A195384...A195385

%C r'2..r'5..r'7...A195386...A195387...A195388...A195389

%C r'3..r'5..r'8...A195395...A195396...A195397...A195398

%C r'7..3....4.....A195399...A195400...A195401...A195402

%C 1....r't..t.....A195403...A195404...A195405...A195406

%C t-1..t....r'3...A195407...A195408...A195409...A195410

%C ...

%C In the special case that P is the incenter, I, each Philo line, being perpendicular to an angle bisector, is constructible, and (A),(B),(C) can be evaluated exactly.

%C For the 3,4,5 right triangle, (A)=(2/3)*sqrt(10), (B)=sqrt(5), (C)=sqrt(8), so that Philo(ABC,I)=((2/3)sqrt(10)+sqrt(5)+sqrt(8))/12, approximately 0.59772335.

%C ...

%C More generally, for arbitrary right triangle (a,b,c) with a<=b<c, let f=2*a*b/(a+b+c). Then, for P=I,

%C (A)=f*sqrt(a^2+(b+c)^2)/(b+c),

%C (B)=f*sqrt(b^2+(c+a)^2)/(c+a),

%C (C)=f*sqrt(2).

%C It appears that I is the only triangle center P for which simple formulas for (A), (B), (C) are available. For P=centroid, see A195304.

%D David Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see chapter 16.

%D Clark Kimberling, Geometry In Action, Key College Publishing, 2003, pages 115-116.

%H Michael Cavers, <a href="http://spikedmath.com/524.html">Spiked Math #524</a> (2012)

%H Clark Kimberling, <a href="/A195284/a195284.pdf">Geometry In Action</a>, 2003, scanned copy (with permission). See pages 115-116.

%F Equals (2/3)*sqrt(10).

%e 2.10818510677891955466592902962...

%p philo := proc(a,b,c) local f, A, B, C, P:

%p f:=2*a*b/(a+b+c):

%p A:=f*sqrt((a^2+(b+c)^2))/(b+c):

%p B:=f*sqrt((b^2+(c+a)^2))/(c+a):

%p C:=f*sqrt(2):

%p P:=(A+B+C)/(a+b+c):

%p print(simplify([A,B,C,P])):

%p print(evalf([A,B,C,P])): end:

%p philo(3,4,5); # _Georg Fischer_, Jul 18 2021

%t a = 3; b = 4; c = 5;

%t h = a (a + c)/(a + b + c); k = a*b/(a + b + c); (* incenter *)

%t f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f1 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (A) 195284 *)

%t f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f2 = (f[t])^(1/2) /. Part[s, 1]

%t RealDigits[%, 10, 100] (* (B) A002163 *)

%t f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f3 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (C) A010466 *)

%t (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)

%o (PARI) (2/3)*sqrt(10) \\ _Michel Marcus_, Dec 24 2017

%Y Cf. A002163, A010466, A195285, A195304.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 14 2011

%E Table and formulas corrected by _Georg Fischer_, Jul 17 2021