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

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

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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.

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