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Dividing the Land Evenly (Posted on 2018-01-09) Difficulty: 2 of 5
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.

See The Solution Submitted by Charlie    
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quick mental take | Comment 1 of 2
1. Take randomly one meridian and its opposite and place yourself in the intersection point of one of those meridians with the Equator, allowing the meridian to divide your body in two similar parts. The great circle constitute by both meridians would place an x% of the land on your right side and (100-x)% at your left side. If you begin walking following the Equator line since when you meet the opposite meridian, at that point the proportions would be inverted. Now you have an x% on your left side and (100-x)% at your right. As the function measuring both proportions in each point of the walk is continuous, at some point you must have had 50%-50% on both sides. The meridian throught that point and its opposite compose a great circle whose all points divides the land in two equal parts.

2. If the Earth is an sphere the precedent way of reasoning do not stand only for meridians. For each great circle it should be posible to determine a pair of points (one and its opposite) in the perpendicular central plane to that circle, so that the great circle perpendicular to that plane, and going throught that pair of points, divides the land in two equal parts. (=this is just applying to a great circle the procedure described for meridians in n. 1).
The intersection of that great circle with the great circle of meridians finded in 1 determine two opposite points having the property of two great circles going throught them and dividing the land in two equal parts.

3. Number 2 proves also 3 as that is true for each great circle that divides the land on two equal amounts of land an whose intersection with the great circle of meridians determined in 1 is variable. 

4. I suppose that in that case the distribution of the land would be simetric or very peculiar with respect the axis of that point and its opposite in the sphere.

Edited on January 9, 2018, 3:32 pm
  Posted by armando on 2018-01-09 10:18:43

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