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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.

  Submitted by Charlie    
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Solution: (Hide)
1. Imagine a movable ring perpendicular to the equator and therefore covering two supplemental meridians. Most likely it will have more land on one side (say the pink side) than the other (say the blue side); if not, you've already got a pair of meridians meeting the criterion. Rotate the ring about the earth's axis. When it's rotated 180° it will be the blue side that has the greater land area. Somewhere in the middle of its rotation it had to switch, at which point it had equal areas of land on either side.

2. Start with a meridian pair found as in part 1. Take a similar ring as was used in it but this time perpendicular to it; the equator might make a good choice. Then start rotating that great circle. In the same way as we proved that the rotating meridian pair will somewhere divide the land as needed, the rotated equator will also do so.

3. Rotate an arbitrary great circle about a diameter far enough as to divide the land areas into two equal parts. Then take a perpendicular to that great circle and rotate it about its own diameter to do the same. The intersections of the two great circles are such a pair of antipodal points. Choice of the original great circle so as to exclude some other pair will preclude these points from being the same points found in another instance.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Idea for Part 4Jer2018-01-09 15:30:46
quick mental takearmando2018-01-09 10:18:43
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