Geometric Proof
That the intersection of a sphere and a plane is, in fact, a circle can be seen as follows. Let the S be a sphere with center O, P a plane which intersects S. Draw OE perpendicular to P and meeting P at E., Let A and B be any two points in the intersection. Then AOE and BOE are right triangles with a common side, OE, and hypotenuses AO and BO, equal. Therefore the remaining sides AE and BE are equal. This proves that all points in the intersection are the same distance from the point E in the plane P, in other words all points in the intersection lie on a circle with center E. Note that OE is the axis of the circle.
As a corollary, on a sphere there is exactly one circle that can be drawn though three given points.
The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.
Read more about this topic: Circle Of A Sphere
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