8 CHAPTER 1 Advanced Euclidean Geometry
This implies in particular that for signed magnitudes,
AB
BA
= − 1.
Before proceeding further, the reader should pay special attention
to the ubiquity of “dropping altitudes” as an auxiliary construction.
Both of the theorems of this subsec-
tion are concerned with the following
configuration: we are given the trian-
gle 4 ABC and points X, Y, and Z on
the lines (BC),(AC),and (AB), respec-
tively. Ceva’s Theorem is concerned with
theconcurrencyof the lines (AX),(BY),
and (CZ). Menelaus’ Theorem is con-
cerned with thecolinearityof the points
X, Y,andZ. Therefore we may regard these theorems as being “dual”
to each other.
In each case, the relevant quantity to consider shall be the product
AZ
ZB
×
BX
XC
×
CY
Y A
Note that each of the factors above is nonnegative precisely when the
pointsX, Y,andZlie on the segments [BC],[AC],and [AB],respec-
tively.
The proof of Ceva’s theorem will be greatly facilitated by the fol-
lowing lemma: