Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2012-04-19 01:06:14

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Maximum common secant line

Here is a question:

Given two circles,which intersect at points A and B,find the angle between the common secant line through A and the chord AB such that the common secant line of the two circles is maximum.The maximum common secant line of two circles is here defined as the line which intersects both circles,such that the portion of that line on the inside of the circles is of maximum length.

We can use GeoGebra to get an idea of what that line could be by looking at a particular case.

1)Draw the points O1(0,0) and O2(2,0).

2)Using the Circle tool draw two circles-c with center O1 and radius 1.5 and d with center O2 and radius 1.

3)In the Input field enter: Intersect[c,d] and name the points of intersection A and B.

4)Make a slider s with range -1.5 to 1.5 .Right-click on the slider,then click Object properties...When a window opens set Increment to 0.001 and Speed to a value between 0.25 and 0.5.

5)Set precision to 10 decimals.

6)In the Input field enter: C=(s,sqrt(2.25-s^2)).

7)Draw the line AB and call it a.

8)Draw the line AC and call it b.

9)Find the intersection of the line b and circle d and call it D.

10)Using the angle tool to get the angle between lines a and b and call that angle alpha.

11)Make a parameter m to watch the distance between points C and D.

12)Now right-click on the slider and click Animation on button.

13)Look at the parameters m and alpha.You should see that m is highest for the values of alpha near 90 degrees.

Now we have an idea of what the angle should be.

A special thanks to the member and administrator bobbym,who started the idea on the forum of showing how we can use GeoGebra for math purposes.

View Image: showimage-2.jpeg

Last edited by anonimnystefy (2012-07-21 13:23:48)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#2 2012-04-19 03:07:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

Hi anonimnystefy;

Very nice work! You have figured geogebra out.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#3 2012-04-19 03:29:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Hi bobbym

Thank you! I came to the idea in helping you with your "mission". Tell me if you find anything that could or should be corrected and/or added. smile


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#4 2012-04-19 03:36:10

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

I am working on it now.

You can leave out the mission idea. Even if that were true I would never tell anyone that.
People with missions end up dead here.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#5 2012-04-19 03:39:52

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Tell me if it works out.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#6 2012-04-19 03:43:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

I have this for you drawing.

View Image: 2012-04-19_204303.gif

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#7 2012-04-19 03:45:40

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

That is good!

I made a picture myself but it wouldn't post hmm

It was a JPG file.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#8 2012-04-19 04:09:14

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

I am getting a maximum closer to 100°


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#9 2012-04-19 04:11:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Let me see.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#10 2012-04-19 08:00:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Could I see what you did to get 100? Could you show me the screenshots in which the angle is close to 90 and and in other to 100°?

Last edited by anonimnystefy (2012-04-19 08:00:43)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#11 2012-04-19 08:04:46

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

Hi;

The problem seems to be with C. The way it is defined you cannot have an |x|>1.5


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#12 2012-04-19 08:20:45

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Did you see the slider range?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#13 2012-04-19 08:31:23

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

Yes, that is the point. That is only half a circle.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#14 2012-04-19 08:36:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

I could add one more point that will go around the other half.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#15 2012-04-19 08:46:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

How about just placing a point C on the larger circle and animating it. No slider needed.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#16 2012-04-19 08:52:51

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

What do you mean?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

#17 2012-04-19 09:04:18

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,429

Re: Maximum common secant line

Try this as an experiment. Use the new point tool to make a point on the circumference of the larger circle. Then right click that point and press animation on. What happened?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#18 2012-04-19 10:04:22

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,812

Re: Maximum common secant line

Okay,I will try it tomorrow! smile


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

Offline

Board footer

Powered by FluxBB