Crossing Circles

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On a plane, you are only allowed to draw circles. After drawing 1 circle, can you ALWAYS draw another so that the new circle crosses all existing circles at 2 points? Why?

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This can be seen by taking a circle of any radius, but lets say one, centered at $(\frac{1}{2},0)$. Then take a copy of that circle, and rotate it $180^ \circ$.

two circles of radius 1, intersecting on the y axis

Then you repeat this process, placing as many circles as you want between them, in the top or bottom of the plane. It could look something like this, which is only 4 circles, but more is harder to tell what is going on.

four circles, all intersecting each other

This can be done as long as they all contain the same interior point and have the same radius. For our example, they all contain the origin.