Harmonic Points and Lines: A Projective Geometric Exercise

WARNING: PROJECTIVE GEOMETRY MAY PUT HOLES IN YOUR BRAIN.  YOUR MIND MIGHT LEAK OUT - IF YOU WANT YOUR MIND IN YOUR BRAIN STOP READING NOW. Okay, here is a geometric exercise that I find very interesting (for some context about WHY it is interesting, look here). It is, however, more complicated than the previous one...

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Why every line is a circle… and vice versa: a projective geometric exercise.

I'd like to make a contribution with regards to circularity/linearity, from a geometrical standpoint.  If you don't like geometry, stop reading, or better yet, read with increased intensity. The polarity between circle/line is one that is fundamental to many geometries - they are taken to be quite different logical entities.  Primarily this arises because of...

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