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A195340 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).

%I A195340 #8 Feb 11 2025 15:21:24
%S A195340 7,8,4,9,2,9,6,8,4,7,6,4,4,3,4,9,4,5,2,0,1,7,2,4,3,6,3,4,5,6,7,3,5,6,
%T A195340 7,2,6,9,4,4,4,6,1,4,2,6,3,0,2,4,6,6,8,1,0,5,5,5,4,4,8,5,5,7,9,2,3,8,
%U A195340 2,7,5,4,4,9,4,8,3,1,3,4,8,5,5,6,0,9,0,1,9,1,7,7,8,0,0,9,1,1,6
%N A195340 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).
%C A195340 See A195284 for definitions and a general discussion.
%H A195340 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%e A195340 (A)=0.7849296847644349452017243634567...
%t A195340 a = 1; b = 2; c = Sqrt[5]; f = 2 a*b/(a + b + c);
%t A195340 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ];
%t A195340 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ];
%t A195340 x3 = f*Sqrt[2];
%t A195340 N[x1, 100]
%t A195340 RealDigits[%] (* (A) A195340 *)
%t A195340 N[x2, 100]
%t A195340 RealDigits[%] (* (B) A195341 *)
%t A195340 N[x3, 100]
%t A195340 RealDigits[%] (* (C) A195342 *)
%t A195340 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195340 RealDigits[%] (* Philo(ABC,I) A195343 *)
%o A195340 (PARI) polrootsreal(x^4 - 520*x^2 + 320)[3] \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195340 Cf. A195284, A195341, A195342, A195343.
%K A195340 nonn,cons
%O A195340 0,1
%A A195340 _Clark Kimberling_, Sep 16 2011