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A195345 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)=(1,3,sqrt(10)).

%I A195345 #7 Feb 11 2025 12:28:55
%S A195345 1,0,3,2,6,4,1,4,9,1,2,0,9,3,4,7,3,6,1,4,7,5,6,3,7,6,5,5,6,5,7,6,1,1,
%T A195345 4,8,5,4,1,4,2,2,0,1,8,5,8,1,6,7,1,1,5,6,8,9,2,7,0,6,5,3,6,1,1,0,5,5,
%U A195345 7,9,5,0,5,3,9,8,2,3,3,0,2,4,9,7,6,0,2,8,0,4,2,2,8,7,6,6,4,8,1,9
%N A195345 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)=(1,3,sqrt(10)).
%C A195345 See A195284 for definitions and a general discussion.
%H A195345 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%e A195345 (B)=1.0326414912093473614756376556576114854142201858...
%t A195345 a = 1; b = 3; c = Sqrt[10]; f = 2 a*b/(a + b + c);
%t A195345 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195345 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195345 x3 = f*Sqrt[2]
%t A195345 N[x1, 100]
%t A195345 RealDigits[%]  (* A195344 *)
%t A195345 N[x2, 100]
%t A195345 RealDigits[%] (* A195345 *)
%t A195345 N[x3, 100]
%t A195345 RealDigits[%] (* A195346 *)
%t A195345 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195345 RealDigits[%] (* A195347 *)
%o A195345 (PARI) (6*sqrt(20 + sqrt(40)))/(14 + 5*sqrt(10)) \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195345 Cf. A195284.
%K A195345 nonn,cons
%O A195345 1,3
%A A195345 _Clark Kimberling_, Sep 16 2011