Theorems of Projective Geometry

A theorem, now considered seminal for projective geometry, was discovered by Pappus of Alexandria in about the year 320  

Calling sides such as 1-4, 2-5, 3-6 opposite, the theorem may be stated in the form  

If a simple hexagon be inscribed in two intersecting lines, the three pairs of opposite sides will intersect in collinear points. 

If A1A4, A2AS, AsAg are concurrent and A2A2, A A, A4A5 are concurrent, then A1A2, A3A4, A Ae are concurrent. 

In given lines a, b, c passing through a point X, and a, b, d passing through a point X set  

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A certain dualism is observable in carbonium ion-carbanion chemistry, a dualism rather like that of lines and points in projective geometry. The reader may recall that interchanging the words "line and "point in a theorem of projective geometry converts it into a statement that is also a theorem, sometimes the same one. For most carbonium ion reactions a corresponding carbanion reaction is known. The dualism can be used as a method for the invention of new, or at least unobserved, carbanion reactions. The carbanionic reaction corresponding to the carbonium ion rearrangement is of course the internal nucleophilic... 

Formal proof of the fundamental theorem, and all other theorems of projective geometry, are given at various levels of rigour in many reference works. Graphical demonstration is condidered adequate for our purposes. 

The same reasoning shows that non-radial planes acquire an extra line at infinity. It follows that projective space may be regarded as affine space plus a plane at infinity. One of the most elegant properties of projective geometry is the principle of duality which asserts that, in a projective plane every definition or theorem remains valid on the consistent interchange of the words line and point. 

Radial lines in projection look like points and radial planes look like lines when viewed edge-on, which means that radial dimensions are lost. This correspondence dehnes the principle of duality which asserts that any dehnition or theorem in projective geometry remains valid on interchanging the words point and line, as well as the operations ... 

In classical projective geometry there is a concept of duality slightly different from the one used so far in this book. It is that every theorem or construction which relates points to lines has a dual which relates lines to points. For example, every pair of distinct points determines a line every pair of distinct lines determines a point. 

Whereas most theorems of geometry are concerned with the concepts distance, angle and congruence, a smaller number of others are only concerned with the incidence of points and straight lines. This distinction differentiates between the common metrical properties of geometry and those, which are independent of measurement, and which reflect the characteristics of what became known as projective geometry. 

This is clearly the same as the theorem of Pappus if, indeed, all conics are equivalent in projective geometry. 

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