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A195304 Decimal expansion of shortest length of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(3,4,5). 64
1, 8, 9, 6, 3, 0, 0, 5, 6, 6, 3, 0, 9, 2, 0, 2, 0, 1, 4, 7, 5, 3, 8, 6, 7, 2, 0, 3, 6, 5, 4, 8, 1, 9, 9, 1, 7, 0, 8, 0, 1, 0, 3, 2, 8, 2, 9, 8, 1, 9, 2, 8, 6, 6, 6, 4, 1, 0, 2, 7, 8, 4, 3, 9, 4, 4, 4, 2, 9, 7, 6, 3, 7, 7, 2, 5, 4, 6, 2, 9, 2, 1, 1, 7, 4, 3, 4, 9, 5, 1, 7, 5, 2, 6, 6, 7, 2, 1, 0, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 sum for ABC and P is here introduced as s=(A)+(B)+(C), and the Philo number for ABC and P, as s/(a+b+c), denoted by Philo(ABC,P).

...

Listed below are examples for which P=G (the centroid); in this list, r'n means sqrt(n) and t=(1+sqrt(5))/2 (the golden ratio).

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

3....4....5......A195304...A195305....A105306...A195411

5....12...13.....A195412...A195413....A195414...A195424

7....24...25.....A195425...A195426....A195427...A195428

8....15...17.....A195429...A195430....A195431...A195432

1....1....r'2....A195433..-1+A179587..A195433...A195436

1....2....r'5....A195434...A195435....A195444...A195445

1....3....r'10...A195446...A195447....A195448...A195449

2....3....r'13...A195450...A195451....A195452...A195453

r'2..r'3..r'5....A195454...A195455....A195456...A195457

1....r'2..r'3....A195471...A195472....A195473...A195474

1....r'3..2......A195475...A195476....A195477...A195478

2....r'5..3......A195479...A195480....A195481...A195482

r'2..r'5..r'7....A195483...A195484....A195485...A195486

r'7..3....4......A195487...A195488....A195489...A195490

1....r't..t......A195491...A195492....A195493...A195494

t-1..t....r'3....A195495...A195496....A195497...A195498

A similar list for P=incenter is given at A195284.

LINKS

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

EXAMPLE

(A)=1.89630056630920201475386720365481991708010328...

MATHEMATICA

a = 3; b = 4; h = 2 a/3; k = b/3;

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) A195304 *)

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]   (* (B) A195305 *)

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]   (* (C) A195306 *)

c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)

RealDigits[%, 10, 100]   (* Philo(ABC, G) A195411 *)

CROSSREFS

Cf. A195305, A195306, A195307; A195284 (P=incenter).

Sequence in context: A093626 A154208 A154491 * A197691 A258104 A253299

Adjacent sequences:  A195301 A195302 A195303 * A195305 A195306 A195307

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 18 2011

STATUS

approved

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Last modified August 18 16:01 EDT 2017. Contains 290727 sequences.