A195304 -id:A195304 - cp's OEIS frontend

A195472 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(B)=1.1324489836725644804259712518338035968298278...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195304.

a = 1; b = Sqrt[2]; 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) A195471 *)
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) A195472 *)
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) A195473 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)

A195473 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(C)=0.8319775602891632045930238114819678...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195304.

a = 1; b = Sqrt[2]; 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) A195471 *)
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) A195472 *)
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) A195473 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)

A195474 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(2),sqrt(3) right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=0.626950112353490925393527524887715891999...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195304.

a = 1; b = Sqrt[2]; 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) A195471 *)
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) A195472 *)
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) A195473 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)

A195479 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(A)=1.24406215675473698925469297613441440690...

Cf. A195304, A195480, A195481, A195482.

a = 2; b = Sqrt[5]; 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) A195479 *)
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) A195480 *)
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) A195481 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195482 *)

A195480 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(B)=1.99586724789639139098163600678265041581157...

Cf. A195304.

a = 2; b = Sqrt[5]; 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) A195479 *)
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) A195480 *)
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) A195481 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195482 *)

A195481 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(C)=1.3569174039377603657928077597670785...

Cf. A195304.

a = 2; b = Sqrt[5]; 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) A195479 *)
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) A195480 *)
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) A195481 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195482 *)

A195482 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,sqrt(5),3 right triangle ABC.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			Philo(ABC,G)=0.635268604839336218811506278276445852019818763...

Cf. A195304.

a = 2; b = Sqrt[5]; 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) A195479 *)
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) A195480 *)
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) A195481 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195482 *)

A195483 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(A)=0.90534709308364721723607657678568...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A195304, A195484, A195485, A195486.

a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)

A195484 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(B)=1.7060463503442324422854199040984706076236802887300...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195304.

a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)

A195485 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 19 2011

See A195304 for definitions and a general discussion.

			(C)=1.294238923692273874334567899656550594640819...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A195304.

a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
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) A195484 *)
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) A195485 *)
c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)

cp's OEIS Frontend

A195472 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).

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A195473 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).

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Mathematica

A195474 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(2),sqrt(3) right triangle ABC.

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A195479 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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A195480 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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Mathematica

A195481 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(2,sqrt(5),3).

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Mathematica

A195482 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,sqrt(5),3 right triangle ABC.

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Mathematica

A195483 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

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Author

Keywords

Comments

Examples

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Programs

Mathematica

A195484 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

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