I have heard that there are infinitely circumferences that passes through two points. I have tried to figure that out. and I thought this may be why.
Let´s see you have two points $(A,B)$, draw a segment between them. Then take the middle and draw a perpendicular that goes through it. Let´s call it "$l$". Then if you take any point $P$ of that parpendicular and unite with both points $A$ and $B$, you will find that $\overline {PA} = \overline {PB}$. You can prove that the pythagoream theorem or with similar triangles. Here is a drawing.
I assume that the center of those infinte circumferences have to be in the perpendicular $l$. But could that not be true? Does it exist a circumference whose center is not there?
Think of your question in a new light. You are given a fixed circle. Then fix a point at the "top" of the circle. How many pairs of points have that one circle passing through them? Surely infinitely many.
Now fix two points, and imagine that they are at opposite ends of a circle. So they describe a chord of $\pi$ radians. Surely, these two points can also trace out a chord of $\pi/2$ radians. The same is true for one of $\pi/3$ radians. And so on.