Let's be very careful here: The "great circle" of a sphere is a circle that lies on the surface of
the sphere, so there's no way the great circle can "pass through" the sphere's center.
However, in order for the circle to be a "great circle", its center must be the center of the sphere.
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Sure.A circle is the shape of every possible CUT through a sphere with a flat knife.
Every cross-section of a sphere is a circle.
It is the center of the circle
A circle has a round shape in which every point on the circle is the same distance from one point called "the center". The center of a circle is not on the circle.
The precise definition of the word 'centres' is the middle point of a circle or a sphere. This means the point is an equal distance from every point on the circle/sphere.