Summary of Geometrical Theorems

Pappus' Theorem: Let three points A, B, C be incident to* a single straight line and another three points a,b,c incident to another straight line.  Then three pairwise intersections 1 = BcbC, 2 = AcaC, and 3 = AbaB are incident to a (third) straight line.  (*See the note under "projective geometry", below, regarding the use of "incident to".)

Brianchon's Hexagon Theorem: in a hexagon circumscribed about a conic, the major diagonals, i. e. the diagonals joining vertices with the opposite ones, are concurrent. (Its dual is Pascal's Theorem)

Pascal's Hexagon Theorem: if a hexagon is inscribed in a conic, then the three points at which the pairs of opposite sides meet lie on a straight line.  (Its dual is Brianchon's Theorem)

 . . . . . . Find all the pages that link to this page, and you'll have a nice list to put here!

 . . . . . . Also, have a look through the "puzzle" pages for statements with proofs.

Projective Geometry

Projective Geometry: A geometry which begins with the ordinary points, lines, and planes of Euclidean plane geometry, and adds an ideal plane, consisting of ideal lines, which, in turn contain ideal points, which are the intersections of parallel lines and planes.

In projective plane geometry, points and lines are considered "duals" of one another.  Two lines always determine (contain) exactly one point; two points always determine exactly one line.

In projective space geometry, points and planes are considered duals of one another.  Three non-collinear points determine a plane; three non-collinear planes determine a point.

The axioms of projective geometry are duals of one another as well, which means the words "point" and "line" can be interchanged in any axiom to get another axiom.  Even more startling is that any proof using these axioms, or derived from other proofs using the axioms can also be changed in the same way to prove its dual.

* To make the terminology easier to convert to its dual terminology, the word "incident to" is used both of a point lying on a line, and of a line passing through a point.  For example, three collinear points are said to be incident to a particular line, and three concurrent lines are said to be incident to a particular point.

Related pages in this website:

Projective Geometry

References

Cut-the-knot: Projective Geometry 

The webmaster and author of the Math Help site is Graeme McRae.
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