Geometry puzzle

the answer is "no"
And you can't solve this with infinitely many circles because you can't have infinity > infinity - 100

Draw x circles such that x is a finite number

Draw a circle such that it touches two circles one time each and encloses all other circles.

Draw a tangent to this circle.

Done.

Now if more circles were added it could be proven that the circle would just get larger and the tangent not intersecting the circles farther away from the initial line satisfying the problem.

Thus, the answer is no, it is not possible.

>Draw x circles such that x is a finite number
finite number of circles is not required.

There are finitely or infinitely many circles on the plane but each line only intersects at most n circles where n is finite.

I actually did exactly what you did but it requires more work to show that you can find such a large circle when there might be infinitely many on the plane.

I'm not sure that x really has to be finite anyway in my proof. The final line of my proof involves me stating that you can increase the size of the overall circles again and again. So we start at a finite number and then end the proof with an increasing amount infinitely. I don't see anywhere you mentioned in your entire proof that the circle enclosing all other circles should touch two of the circles. That's important because it describes how to draw said circle.

The point was, you wanted a proof which was easier and shorter, and mine is valid, easier, and shorter. You can have all the fancy notation you want, mine is fundamentally correct and you have not provided any satisfactory objection to it yet.

Your's is invalid; it assumes the union of all the circles is bounded. This isn't necessarily true with infinite circles.

It is impossible to draw or conceive of an infinite shape.

The point is invalid, again.

To make things easier for you to understand, and infinite circle would really just resolve itself into being an equiangular hyperbola, if you tried to manifest it.

show me your big, all encompassing circle here (the tiling continues indefinetly)
>b..but there are obviously lines which intersect infinitely many lines
What if I found a complex infinite tiling which manages to solve the problem?

Did you not read the proof. If we start from the supposition of infinite circles, it cannot be completed. You are making a mistake of presupposing I start at this notion.

If I start at a finite number of circles, then the solution to the proof can be expanded to an infinite number of circles.

that's not how proofs work.
You are proving the problem for any finite number of circles.
But any finitee number is not equivalent to infinite number.
I will show you the equivalent of your logic with series:
for any sum of natural numbers up to some n we can find an upper bound, i.e. 1+2+3 < 10
1+2+3+4+5+6+7 < 100 etc.
Therefore, the infinite sum of all natural numbers is bounded.
This is what you are doing.