Two circle intersection

Question

Two circles intersect at two points maximum when we want to draw intersecting circles. But there we are solving quadratic equations, what is the argument about the other two missing points?

Answer 1

Suppose the centers of the circles are $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$. Notice that-

1. The line joining the intersection points must be perpendicular to the line joining the points $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$.

2. The line joining $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$ must also pass through the midpoint of the line segment joining the intersection points.

There are at most two distinct points satisfying the two conditions.