20 October 2013

Two projective theorems

Projective geometry is a beautiful branch of mathematics, in which straight angles, parallel lines and circles are conspicuously lacking—simpler and more basic than 'usual' geometry. Here is a nice example of a projective property. Simple, but unexpected and not the least intuive. Who would expect something invariant about this configuration!

Theorem. If four concurrent rays are given (yellow), and a transversal (white, 3 examples) intersects them in the points A,B,C,D respectively, then (AC x BD) : (AD x BC) is independent of the position of the transversal.
 Here (also here) the straightforward (if analytical) verification. Actually, there is more to it than what we give here; the real invariant also takes the order of the points into account, and results in an invariant (called cross-ratio) which can also be negative.

Now for a more sophisticated example.

Theorem. Consider three concurrent lines (white). Take two points A,D on the first and two points B,C on the third such that the intersection X of AD and BC is on the second. Let E be the intersection of the straight line through A,C and the straight line through B,D. Whatever the position of A,B,C,D, the point E will remain on the same straight line, concurrent with the given lines. (Three configurations, in different colours, drawn. The resulting line is dashed.)
Proof by brute force is possible, but not recommended. Projective geometry has its own tools (which would reveal that any transversal yields four intersection points with cross-ratio equal to -1).