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A195348 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,sqrt(3),2) and vertex angles of degree measure 30,60,90.

%I A195348 #5 Mar 30 2012 18:57:45
%S A195348 7,5,7,8,7,4,7,6,3,9,2,6,0,2,3,9,9,8,8,1,2,1,8,6,7,4,7,4,2,7,0,0,9,5,
%T A195348 3,0,3,4,6,7,9,2,5,4,0,1,9,4,4,5,2,0,3,5,8,4,1,3,3,3,8,1,7,4,6,1,0,0,
%U A195348 9,1,5,8,9,3,3,7,9,8,1,0,2,3,2,1,8,3,1,2,7,1,1,0,1,2,8,5,8,2,1,3
%N A195348 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,sqrt(3),2) and vertex angles of degree measure 30,60,90.
%C A195348 See A195284 for definitions and a general discussion.
%e A195348 (A)=0.7578747639260239988121867474270095303467925401944...
%e A195348 (A)=(4*sqrt(6-3*sqrt(3)))/(3+sqrt(3))
%e A195348 (B)=2-(2/3)sqrt(3)
%e A195348 (C)=sqrt(6)-sqrt(2)
%t A195348 a = 1; b = Sqrt[3]; c = 2;
%t A195348 f = 2 a*b/(a + b + c);
%t A195348 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195348 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195348 x3 = f*Sqrt[2]
%t A195348 N[x1, 100]
%t A195348 RealDigits[%] (* (A) A195348 *)
%t A195348 N[x2, 100]
%t A195348 RealDigits[%] (* (B) A093821 *)
%t A195348 N[x3, 100]
%t A195348 RealDigits[%] (* (C) A120683 *)
%t A195348 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195348 RealDigits[%] (* A195380 *)
%Y A195348 Cf. A195284, A093821, A120683, A195380.
%K A195348 nonn,cons
%O A195348 0,1
%A A195348 _Clark Kimberling_, Sep 17 2011