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A195341 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).

%I A195341 #13 Feb 11 2025 15:22:50
%S A195341 8,9,8,0,5,5,9,5,3,1,5,9,1,7,0,7,4,4,8,8,3,8,9,0,3,0,3,5,9,5,0,5,3,5,
%T A195341 7,5,1,5,8,4,2,4,9,6,4,6,4,2,1,6,7,0,3,8,7,8,8,9,5,7,2,6,8,3,3,7,6,1,
%U A195341 5,1,1,2,9,5,8,0,4,1,6,5,3,1,9,2,8,6,1,1,5,5,0,1,8,6,8,5,1,6,0,9,7,6,8
%N A195341 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).
%C A195341 See A195284 for definitions and a general discussion.
%H A195341 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%e A195341 0.898055953159170744883890303595053575158424964642...
%t A195341 a = 1; b = 2; c = Sqrt[5]; f = 2 a*b/(a + b + c);
%t A195341 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ];
%t A195341 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ];
%t A195341 x3 = f*Sqrt[2];
%t A195341 N[x1, 100]
%t A195341 RealDigits[%] (* (A) A195340 *)
%t A195341 N[x2, 100]
%t A195341 RealDigits[%] (* (B) A195341 *)
%t A195341 N[x3, 100]
%t A195341 RealDigits[%] (* (C) A195342 *)
%t A195341 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195341 RealDigits[%] (* Philo(ABC,I) A195343 *)
%o A195341 (PARI) polrootsreal(x^4 - 100*x^2 + 80)[3] \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195341 Cf. A195284.
%K A195341 nonn,cons
%O A195341 0,1
%A A195341 _Clark Kimberling_, Sep 16 2011
%E A195341 a(99)-a(100) corrected by _Georg Fischer_, Jul 18 2021