Projective Geometry -- from Wolfram MathWorld: "The most amazing result arising in projective geometry is the duality principle, which states that a duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other. More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line."
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