Prove that a point in a circle equidistant from any three points on the circle is the centre

Question

1. Prove that a point in a circle equidistant from any three points on the circle is the centre

2. $\frac{1-1}{1-1}=?$

Answer 1

Assume that $p$ is a point in the plane that is equidistant (say of length $r$) to three distinct points on the circle. Then the circle at $p$ of radius $r$ intersect the original circle at three points. If two circles intersect at three points then they must be the same and hence $p$ is the center of the original circle.