TITLE: A sphere that contains a circle and a point
QUESTION [7 upvotes]: How to prove that a circle and a point outside the plane of the circle determine a sphere?
I know that the circle is determined by three non-collinear points, so from the circle, we get 3 non-collinear points and we also have an extra point which is outside the plane of the circle, therefore, we have 4 non-coplanar points. These will determine a sphere.
But how do we prove that the sphere that we formed contains all of the circle from before?

REPLY [3 votes]: Note that three non-collinear points determine both a plane and a unique circle in that plane. Intersecting the plane with a sphere containing the the three points gives a circle (intersection of plane with sphere) containing those three points, which must be the same circle (because it was unique).