The theorems of Menelaus and Ceva

Triangle geometry is inexhaustible. Among its countless riches, two results of similar form stand out: Menelaus's and Ceva's theorems. Start with a triangle ABC. Let P lie on (BC), Q on (CA), and R on (AB). We will also use directed lengths, so that any segment UV can be assigned a sign. The directed length $\overline{\text{UV}}\{\backslash overline\; \{\backslash text\{UV\}\}\}$ of the segment [UV] is equal to its length when measured from U to V, and to the negative of its length when measured from V to U. That is all we need to state Menelaus's and Ceva's theorems!

M

enelaus's theorem: the points P, Q and R are collinear if and only if