1. ## Disjoint Circles

How can prove that?
Let C be a collection of disjoint circles in the plane. Show that there is a line that passes through the origin and not any of these tangent circles.

THANKS!

2. Originally Posted by rhemo
How can prove that?
Let C be a collection of disjoint circles in the plane. Show that there is a line that passes through the origin and not any of these tangent circles.

THANKS!
I don't see why this is true. Why not the infinite collection of circle centered on $O$ with radius $\frac{1}{n},\text{ }n\in\mathbb{N}$?

3. Again:

Let C a collection of disjoint circles in the plane. Show that there is a line that passes through the origin and is not tangent to any of these circles

(Excuse-me my bad English, I don't speak that language)

4. Originally Posted by rhemo
How can prove that?
Let C be a collection of disjoint circles in the plane. Show that there is a line that passes through the origin and not any of these tangent circles.
I think that by disjoint circles the OP means that to circles share neither a point nor an interior point, in the sense of disks.
If that is the case, then the collection is countable. WHY?
If a circle in the plane does not contain the origin as a point or interior point then there is at least one and at most two tangents to the circle that contain the origin.
There are uncountably many lines through the origin.
Can you finish?

5. Originally Posted by Plato
I think that by disjoint circles the OP means that to circles share neither a point nor an interior point, in the sense of disks.
If that is the case, then the collection is countable. WHY?
If a circle in the plane does not contain the origin as a point or interior point then there is at least one and at most two tangents to the circle that contain the origin.
There are uncountably many lines through the origin.
Can you finish?
Clearly, each of this circles can be uniquely identified with a unique point of the rationals, thus it is countable. But, I have never heard it meant that way before? I thought usually disjoint meant what you said, and concentric.

6. Originally Posted by Drexel28
But, I have never heard it meant that way before? I thought usually disjoint meant what you said, and concentric.
The poster said he/she is not an Enghish speaker.
Just replace the word circle with the word closed disk.

7. Originally Posted by Plato
The poster said he/she is not an Enghish speaker.
Just replace the word circle with the word closed disk.
That makes sense

8. Thanks a lot!!!