A question about circles

[reels22]

The definition of a circle specifies that the points that are equidistant from a
given point (the center) are “in a plane.” Why? What would happen if the
points were not restricted to a plane?

 

[R.M.]

If the points were not restricted a plane, then all the points that are equidistant from a given point would define a sphere (3-D). A circle is 2-D, so the plane restriction is necessary.

 

[HallsofIvy]

Or, in four or more dimensions, a "hyper-sphere"!