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A195284 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.

%I A195284 #43 Jan 29 2025 14:58:35
%S A195284 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 A195284 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 A195284 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 A195284 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 A195284 Apart from the first digit, the same as A176219 (decimal expansion of 2+2*sqrt(10)/3).
%C A195284 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 A195284 ...
%C A195284 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 A195284 ...
%C A195284 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 A195284 a....b....c.......(A).......(B).......(C)....Philo(ABC,I)
%C A195284 3....4....5.....A195284...A002163...A010466...A195285
%C A195284 5....12...13....A195286...A195288...A010487...A195289
%C A195284 7....24...25....A195290...A010524...15/2......A195292
%C A195284 8....15...17....A195293...A195296...A010524...A195297
%C A195284 28...45...53....A195298...A195299...A010466...A195300
%C A195284 1....1....r'2...A195301...A195301...A163960...A195303
%C A195284 1....2....r'5...A195340...A195341...A195342...A195343
%C A195284 1....3....r'10..A195344...A195345...A195346...A195347
%C A195284 2....3....r'13..A195355...A195356...A195357...A195358
%C A195284 2....5....r'29..A195359...A195360...A195361...A195362
%C A195284 r'2..r'3..r'5...A195365...A195366...A195367...A195368
%C A195284 1....r'2..r'3...A195369...A195370...A195371...A195372
%C A195284 1....r'3..2.....A195348...A093821...A120683...A195380
%C A195284 2....r'5..3.....A195381...A195383...A195384...A195385
%C A195284 r'2..r'5..r'7...A195386...A195387...A195388...A195389
%C A195284 r'3..r'5..r'8...A195395...A195396...A195397...A195398
%C A195284 r'7..3....4.....A195399...A195400...A195401...A195402
%C A195284 1....r't..t.....A195403...A195404...A195405...A195406
%C A195284 t-1..t....r'3...A195407...A195408...A195409...A195410
%C A195284 ...
%C A195284 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 A195284 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 A195284 ...
%C A195284 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 A195284 (A)=f*sqrt(a^2+(b+c)^2)/(b+c),
%C A195284 (B)=f*sqrt(b^2+(c+a)^2)/(c+a),
%C A195284 (C)=f*sqrt(2).
%C A195284 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 A195284 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 A195284 Clark Kimberling, Geometry In Action, Key College Publishing, 2003, pages 115-116.
%H A195284 Michael Cavers, <a href="http://spikedmath.com/524.html">Spiked Math #524</a> (2012)
%H A195284 Clark Kimberling, <a href="/A195284/a195284.pdf">Geometry In Action</a>, 2003, scanned copy (with permission). See pages 115-116.
%H A195284 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%F A195284 Equals (2/3)*sqrt(10).
%e A195284 2.10818510677891955466592902962...
%p A195284 philo := proc(a,b,c) local f, A, B, C, P:
%p A195284 f:=2*a*b/(a+b+c):
%p A195284 A:=f*sqrt((a^2+(b+c)^2))/(b+c):
%p A195284 B:=f*sqrt((b^2+(c+a)^2))/(c+a):
%p A195284 C:=f*sqrt(2):
%p A195284 P:=(A+B+C)/(a+b+c):
%p A195284 print(simplify([A,B,C,P])):
%p A195284 print(evalf([A,B,C,P])): end:
%p A195284 philo(3,4,5); # _Georg Fischer_, Jul 18 2021
%t A195284 a = 3; b = 4; c = 5;
%t A195284 h = a (a + c)/(a + b + c); k = a*b/(a + b + c); (* incenter *)
%t A195284 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195284 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195284 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195284 RealDigits[%, 10, 100] (* (A) 195284 *)
%t A195284 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195284 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195284 f2 = (f[t])^(1/2) /. Part[s, 1]
%t A195284 RealDigits[%, 10, 100] (* (B) A002163 *)
%t A195284 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195284 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195284 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195284 RealDigits[%, 10, 100] (* (C) A010466 *)
%t A195284 (f1 + f2 + f3)/(a + b + c)
%t A195284 RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)
%o A195284 (PARI) (2/3)*sqrt(10) \\ _Michel Marcus_, Dec 24 2017
%Y A195284 Cf. A002163, A010466, A195285, A195304.
%K A195284 nonn,cons
%O A195284 1,1
%A A195284 _Clark Kimberling_, Sep 14 2011
%E A195284 Table and formulas corrected by _Georg Fischer_, Jul 17 2021