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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

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Last modified March 17 06:34 EDT 2018. Contains 300543 sequences. (Running on oeis4.)