A195298 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)=(28,45,53).

2, 0, 8, 0, 0, 3, 1, 3, 9, 6, 9, 3, 7, 2, 9, 0, 9, 3, 4, 5, 9, 9, 2, 2, 9, 2, 8, 3, 2, 9, 3, 4, 3, 7, 9, 4, 1, 0, 7, 9, 3, 3, 4, 1, 9, 5, 0, 2, 1, 8, 5, 0, 6, 9, 6, 6, 7, 9, 4, 8, 0, 5, 1, 1, 7, 9, 5, 4, 6, 1, 6, 3, 9, 6, 0, 7, 1, 1, 5, 7, 6, 6, 6, 6, 5, 5, 9, 4, 1, 1, 6, 8, 8, 0, 2, 6, 4, 7, 8

Offset: 2

Author

Clark Kimberling, Sep 14 2011

Keywords

• nonn
• cons

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=20.800313969372909345992292832934379410...
		

Crossrefs

Cf. A195284, A195299, A195300, A195301.

Programs

Mathematica

a = 28; b = 45; c = 53;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195298 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (B) A195299 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C)=20*sqrt(2) *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)