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A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

%I A195304 #10 Feb 03 2025 09:07:03
%S A195304 1,8,9,6,3,0,0,5,6,6,3,0,9,2,0,2,0,1,4,7,5,3,8,6,7,2,0,3,6,5,4,8,1,9,
%T A195304 9,1,7,0,8,0,1,0,3,2,8,2,9,8,1,9,2,8,6,6,6,4,1,0,2,7,8,4,3,9,4,4,4,2,
%U A195304 9,7,6,3,7,7,2,5,4,6,2,9,2,1,1,7,4,3,4,9,5,1,7,5,2,6,6,7,2,1,0,7
%N A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5).
%C A195304 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 A195304 ...
%C A195304 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).
%C A195304 ...
%C A195304 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).
%C A195304 a....b...c........(A).......(B)........(C)...Philo(ABC,G)
%C A195304 3....4....5......A195304...A195305....A105306...A195411
%C A195304 5....12...13.....A195412...A195413....A195414...A195424
%C A195304 7....24...25.....A195425...A195426....A195427...A195428
%C A195304 8....15...17.....A195429...A195430....A195431...A195432
%C A195304 1....1....r'2....A195433..-1+A179587..A195433...A195436
%C A195304 1....2....r'5....A195434...A195435....A195444...A195445
%C A195304 1....3....r'10...A195446...A195447....A195448...A195449
%C A195304 2....3....r'13...A195450...A195451....A195452...A195453
%C A195304 r'2..r'3..r'5....A195454...A195455....A195456...A195457
%C A195304 1....r'2..r'3....A195471...A195472....A195473...A195474
%C A195304 1....r'3..2......A195475...A195476....A195477...A195478
%C A195304 2....r'5..3......A195479...A195480....A195481...A195482
%C A195304 r'2..r'5..r'7....A195483...A195484....A195485...A195486
%C A195304 r'7..3....4......A195487...A195488....A195489...A195490
%C A195304 1....r't..t......A195491...A195492....A195493...A195494
%C A195304 t-1..t....r'3....A195495...A195496....A195497...A195498
%C A195304 A similar list for P=incenter is given at A195284.
%H A195304 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>
%e A195304 1.89630056630920201475386720365481991708010328....
%t A195304 a = 3; b = 4; h = 2 a/3; k = b/3;
%t A195304 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195304 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195304 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195304 RealDigits[%, 10, 100]   (* (A) A195304 *)
%t A195304 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195304 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195304 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195304 RealDigits[%, 10, 100]   (* (B) A195305 *)
%t A195304 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195304 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195304 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195304 RealDigits[%, 10, 100]   (* (C) A195306 *)
%t A195304 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195304 RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)
%o A195304 (PARI) polrootsreal(2025*x^6 + 21429*x^4 + 4939*x^2 - 389017)[2] \\ _Charles R Greathouse IV_, Feb 03 2025
%Y A195304 Cf. A195305, A195306, A195307; A195284 (P=incenter).
%K A195304 nonn,cons
%O A195304 1,2
%A A195304 _Clark Kimberling_, Sep 18 2011