Advanced High-School Mathematics

20 CHAPTER 1 Advanced Euclidean Geometry

Step 1. Locate the pointGon the lines (AE) and (FB); we shall

analyze the triangle 4 GHIas indicated below.^4

Step 2. Look at the transversals, applying Menelaus’ theorem to

each:

(^4) Of course, it may be that (AE) and (FB) are parallel. In fact, it may happen that all analogous
choices for pairs of lines are parallel, which would render the present theme invalid. However, while
the present approach uses Menelaus’ theorem, which is based on “metrical” ideas, Pappus’ theorem
is a theorem only about incidence and colinearity, making it really a theorem belonging to “projective
geometry.” As such, if the lines (AE) and (BF) were parallel, then projectively they would meet
“at infinity;” one could then apply a projective transformation to move this point at infinity to the
finite plane, preserving the colinearity ofX,Y, andZ