As the real projective plane, PG(2, R), is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:

Desargues' theorem ⇔ Converse of Desargues' theorem
Pascal's theorem ⇔ Brianchon's theorem
Menelaus' theorem ⇔ Ceva's theorem

I get some dual theorem in wiki. So besides the above theorem, is there any other theorem about duality in projective space? Thank you.


1 Answer 1


A century or more ago, projective geometry texts were often written in a two column format, where theorems and their duals (and their respective definitions and proofs) were described side by side.

For example Hatton's Projective Geometry. The URL I've given points to a sequence of pages written in this format, but if you leaf and skip through the text you'll see many more pages like this, starting all the way back to the beginning of the text.

Back in the day, a dual theorem was called "a correlative". You'll see that term often. This text, like many others of its day, is an all-you-can-eat buffet of dual theorems.

Since you mentioned Pascal $\iff$ Brianchon, see Chapter XIV for that and two other biggies: Carnot's Theorem and its correlative, and Desargues' Theorem (the 'other' Desargues' theorem about transversals through systems of conics) and its correlative).


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