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

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

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

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.