cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

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

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

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).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

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 *)

A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Sep 14 2011

Keywords

Comments

See A195284 for definitions and a general discussion. This constant is the maximum of Philo(ABC,I) over all triangles ABC.

Examples

			Philo(ABC,I)=0.614058971032212611546384892539385408260...

Links

Crossrefs

Cf. A195284.

Programs

  • Mathematica
    a = 1; b = 1; c = Sqrt[2];
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, 1]
RealDigits[%, 10, 100] (* (A) A195301 *)
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, 4]
RealDigits[%, 10, 100] (* (B)=(A) *)
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, 1]
RealDigits[%, 10, 100] (* (C) A163960 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100]  (* Philo(ABC,I), A195303 *)
  • PARI
    (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))) \\ Michel Marcus, Jul 27 2018

Formula

Equals (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))).
Showing 1-3 of 3 results.

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