|
Let A, B, and C be the three
vertices of a triangle. The extension of line segment AB is defined
to be the line, coinciding with the segment, but extending beyond A
and B to infinity. Let F be a point on the extension of segment AB.
(Note that this point can be between A and B, but is not required to
be.), Similarly, let E be on the extension of AC, and let D be on the
extension of BC. Menelaus proved that points D, E, and F can only be
collinear (in other words, fall on a straight line) if the lengths of
the line segments AF X BD X CE = BF X CD X AE.
Menelaus presented this finding
with his own geometric proof, around 100AD.
For all three of the projective
geometry theorems discussed here, one of the most important applications
was in the subsequent development of perspective theory, and its application
to, for instance, technical drawing.
|