SECTION 1.2 Triangle Geometry 11
the triangle. When these three points are collinear, the line formed
is called atransversal. The reader can quickly convince herself that
there are two configurations related to 4 ABC:
As with Ceva’s theorem, the relevant quantity is the product of the
sensed ratios:
AZ
ZB
·
BX
XC
·
CY
Y A
;
in this case, however, we see that either one or three of the ratios must
be negative, corresponding to the two figures given above.
Menelaus’ Theorem. Given the triangle 4 ABC and given points
X, Y, and Z on the lines (BC),(AC), and (AB), respectively, then
X, Y, andZ are collinear if and only if
AZ
ZB
×
BX
XC
×
CY
Y A
=− 1.
Proof. As indicated above, there are two cases to consider. The first
case is that in which two of the pointsX, Y, orZare on the triangle’s
sides, and the second is that in which none ofX, Y, or Z are on the
triangle’s sides. The proofs of these cases are formally identical, but
for clarity’s sake we consider them separately.