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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. 94
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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...A010466...A002163...A195285

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

7....24...25....A195290...A010524.... 7.5 ....A195292

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

8....5....53....A195298... 20*r'2 ..A195299...A195300

1....1....r'2...A195301...A193960...A195301...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) are 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(8)+sqrt(5))/12,

approximately 0.59772335.

...

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

a<=b<c, let f=2ab/(a+b+c).  Then, for P=I,

(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.

REFERENCES

Clark Kimberling, Geometry In Action, Key College Publishing, 2003, pages 115-116.

LINKS

Table of n, a(n) for n=1..100.

Michael Cavers, Spiked Math #524 (2012)

Clark Kimberling, Geometry In Action, 2003, scanned copy (with permission). See pages 115-116.

EXAMPLE

(A)=2.10818510677891955466592902962...=(2/3)sqrt(10).

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 *)

CROSSREFS

Cf. A195285, A010466, A002163, A195304.

Sequence in context: A188835 A217735 A075615 * A076341 A110510 A051122

Adjacent sequences:  A195281 A195282 A195283 * A195285 A195286 A195287

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 14 2011

STATUS

approved

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Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.