Among one branch of numerology, Pyramidology, is the notion that the great circle consisting of the meridian of the Great pyramid of Egypt and its antipodal meridian (east and west longitudes add to 180°), divides the earth's land area into two equal parts (and consequently the water area also, as by definition it divides the whole sphere into two equal parts).

Regardless of the veracity of this claim:

1. Prove that some such great circle consisting of two opposite meridians must have that attribute.

2. Prove that some point and its antipodal point have two great circles going through it (not necessarily either of them being coincidental with meridians) that divide the land area into two equal parts.

3. Prove that there are more than one point (or rather more than one pair of antipodal points) meeting the criterion of part 2.

4. Must there be a point on the earth's surface where all great circles passing through it divide the earth's land area into two equal parts? If so prove it. If not, give a proof that it's highly unlikely to happen by chance.

Assume the earth is a perfect sphere, rather than oblate.