A195284 Decimal expansion of shortest length of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5); i.e., decimal expansion of 2*sqrt(10)/3.

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

Offset: 1

Author

Clark Kimberling, Sep 14 2011

Keywords

• nonn
• cons

Comments

Apart from the first digit, the same as A176219 (decimal expansion of 2+2*sqrt(10)/3).

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P. Philo lines are not generally Euclidean-constructible.

...

Suppose that P lies inside a triangle ABC. Let (A) denote the shortest length of segment from AB through P to AC, and likewise for (B) and (C). The Philo number for ABC and P is here introduced as the normalized sum ((A)+(B)+(C))/(a+b+c), denoted by Philo(ABC,P).

...

Listed below are examples for which P=incenter (the center, I, of the circle inscribed in ABC, the intersection of the angle bisectors of ABC); in this list, r'x means sqrt(x), and t=(1+sqrt(5))/2 (the golden ratio).

a....b....c.......(A).......(B).......(C)....Philo(ABC,I)

3....4....5.....A195284...A002163...A010466...A195285

5....12...13....A195286...A195288...A010487...A195289

7....24...25....A195290...A010524...15/2......A195292

8....15...17....A195293...A195296...A010524...A195297

28...45...53....A195298...A195299...A010466...A195300

1....1....r'2...A195301...A195301...A163960...A195303

1....2....r'5...A195340...A195341...A195342...A195343

1....3....r'10..A195344...A195345...A195346...A195347

2....3....r'13..A195355...A195356...A195357...A195358

2....5....r'29..A195359...A195360...A195361...A195362

r'2..r'3..r'5...A195365...A195366...A195367...A195368

1....r'2..r'3...A195369...A195370...A195371...A195372

1....r'3..2.....A195348...A093821...A120683...A195380

2....r'5..3.....A195381...A195383...A195384...A195385

r'2..r'5..r'7...A195386...A195387...A195388...A195389

r'3..r'5..r'8...A195395...A195396...A195397...A195398

r'7..3....4.....A195399...A195400...A195401...A195402

1....r't..t.....A195403...A195404...A195405...A195406

t-1..t....r'3...A195407...A195408...A195409...A195410

...

In the special case that P is the incenter, I, each Philo line, being perpendicular to an angle bisector, is constructible, and (A),(B),(C) can be evaluated exactly.

For the 3,4,5 right triangle, (A)=(2/3)*sqrt(10), (B)=sqrt(5), (C)=sqrt(8), so that Philo(ABC,I)=((2/3)sqrt(10)+sqrt(5)+sqrt(8))/12, approximately 0.59772335.

...

More generally, for arbitrary right triangle (a,b,c) with a<=b

(A)=f*sqrt(a^2+(b+c)^2)/(b+c),

(B)=f*sqrt(b^2+(c+a)^2)/(c+a),

(C)=f*sqrt(2).

It appears that I is the only triangle center P for which simple formulas for (A), (B), (C) are available. For P=centroid, see A195304.

Examples

			2.10818510677891955466592902962...
		

References

• David Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see chapter 16.
• Clark Kimberling, Geometry In Action, Key College Publishing, 2003, pages 115-116.

Links

• Michael Cavers, Spiked Math #524 (2012)
• Clark Kimberling, Geometry In Action, 2003, scanned copy (with permission). See pages 115-116.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Cf. A002163, A010466, A195285, A195304.

Programs

Maple

philo := proc(a,b,c) local f, A, B, C, P:
f:=2*a*b/(a+b+c):
A:=f*sqrt((a^2+(b+c)^2))/(b+c):
B:=f*sqrt((b^2+(c+a)^2))/(c+a):
C:=f*sqrt(2):
P:=(A+B+C)/(a+b+c):
print(simplify([A,B,C,P])):
print(evalf([A,B,C,P])): end:
philo(3,4,5); # Georg Fischer, Jul 18 2021

Mathematica

a = 3; b = 4; c = 5;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c); (* incenter *)
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) 195284 *)
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]
f2 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (B) A002163 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A010466 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,I) A195285 *)

PARI

(2/3)*sqrt(10) \\ Michel Marcus, Dec 24 2017

Formula

Equals (2/3)*sqrt(10).

Extensions

Table and formulas corrected by Georg Fischer, Jul 17 2021

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