In France, Thales' theorem concerns parallel lines and proportionality. Its proof, combining algebra and plane geometry, is remarkably elegant. It also leads naturally to two landmark results in triangle geometry: Menelaus' and Ceva's theorems.
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On his appointment as a professor at the University of Erlangen in 1872, Felix Klein, then only 23, presented a research program in geometry that has since become known as the "Erlangen Program." The concept of a group lies at its heart.
Menelaus's and Ceva's theorems, two classics of plane geometry, are similar in form. This resemblance becomes clearer when the notion of cross-ratio is introduced.
The idea of pairing two mathematical objects according to a set of rules allows us to extend certain results without reinventing everything: that is the whole point of duality. Or, mathematically speaking, how to kill two birds with one stone.
Points and lines in the plane are dual notions: theorems about collinear points correspond to theorems about concurrent lines. This duality can be defined geometrically. It even extends to space, through coplanarity.
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