Neat question! Earth has a unique equator
because of its shape. You could cut it in an
infinite number of ways with the cut going
through the center of the earth. Any way that
you cut it, there would be a line around the
sphere where the cut intersects its surface;
this line is an equator. If earth was a sphere,
all of these lines would be the same length, and
you could call any one of them the equator.
However… earth is not a perfect sphere; it is an
ellipsoid (a squashed sphere; see the attached
figure). It sort of bulges out in the middle, so
it is widest at the equator. If you cut the
earth with a bunch of different planes that go
through the center, the equator would be the
longest line on the surface.
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