Calculating circle equation given 2 points on it

Question

Let's assume we have 2 points which are on the one base unit circle. There is another circle which passes through those points. How to determine circle equation given coordinates for those points. You can see what I mean in the image below

Answer 1

There's no way to determine a circle just by two points, you need minimum of 3 points to describe a circle uniquely.

Answer 2

You cannot do this. 2 points are not enough to define a circle. There are infinitely many circles passing though these 2 points.

Why? Imagine your points are $A$ and $B$. Draw a line $L$ through the middle of $AB$ which is perpendicular to $AB$. Now, if you pick any point $M$ from $L$, you can draw a circle with center $M$ which passes through the points $A$ and $B$. Since you have infinitely many points $M$ on $L$, it means you have infinitely many circles through $A$ and $B$.