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A195399 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)=(sqrt(7),3,4).

%I A195399 #12 Nov 26 2024 16:30:03
%S A195399 1,7,5,9,3,8,2,1,5,7,0,9,6,4,9,2,5,5,8,8,6,5,1,6,3,5,2,4,9,0,0,3,8,2,
%T A195399 0,7,0,9,2,3,3,3,8,0,9,1,3,8,8,5,4,5,5,9,0,2,6,6,5,7,5,0,5,6,7,4,7,1,
%U A195399 6,9,1,9,7,9,7,9,3,7,4,3,5,5,4,2,1,6,8,6,5,2,7,1,7,1,1,7,4,6,9,6
%N A195399 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)=(sqrt(7),3,4).
%C A195399 See A195284 for definitions and a general discussion.
%H A195399 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%e A195399 (A)=1.75938215709649255886516352490038207092333...
%t A195399 a = Sqrt[7]; b = 3; c = 4;
%t A195399 f = 2 a*b/(a + b + c);
%t A195399 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195399 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195399 x3 = f*Sqrt[2]
%t A195399 N[x1, 100]
%t A195399 RealDigits[%]    (* (A) A195399 *)
%t A195399 N[x2, 100]
%t A195399 RealDigits[%]    (* (B) A195400 *)
%t A195399 N[x3, 100]
%t A195399 RealDigits[%]    (* (C) A195401 *)
%t A195399 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195399 RealDigits[%]    (*  Philo(ABC,I) A195402 *)
%o A195399 (PARI) sqrt(8)*(7-sqrt(7))/7 \\ _Charles R Greathouse IV_, Nov 26 2024
%o A195399 (PARI) polrootsreal(49*x^4-896*x^2+2304)[3] \\ _Charles R Greathouse IV_, Nov 26 2024
%Y A195399 Cf. A195284, A195400, A195401, A195402.
%K A195399 nonn,cons
%O A195399 1,2
%A A195399 _Clark Kimberling_, Sep 17 2011