Earth's centre point lies exactly in The Holy Ka'aba, Mecca, Saudi Arabia.
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That's a sphere.
If ... the square of (the x-coordinate of the point minus the x-coordinate of the center of the circle) added to the square of (the y-coordinate of the point minus the y-coordinate of the center of the circle) is equal to the square of the circle's radius, then the point is on the circle.
No, it is a point on the earth's surface, and thus cannot be the centre as well.
There are infinitely many such points. One of them is: (2,236, 4.291)
There are infinitely many points. One of these is (10, 0).