In the projective geometry pages, we met the
The Inscribed Quadrilateral Theorem
If the vertices of a quadrilateral lie on a pconic, then
the line joining the intersections of opposite sides
is the polar of the intersection of the lines joining
opposite vertices.
In the sketch on the right, the pline QR is the polar of P.
Suppose that the chord AC meets QR at S. Then the above
polarchord theorem shows that (A,C,P,S) = 1.
As always, the picture shows an embedding of the projective
figure  one in which none of the ppoints embeds as an ideal
point. In the embedding, we can interpret the crossratio in
terms of signed ratios (see the embedding theorem).
Thus (AP/PC)/(AS/SC) = 1, so that AP/PC =  AS/SC.
We can obtain this last equality in another way :
Applying Ceva's Theorem to ΔABC and the point D, we get
(AP/PC)(CR/RB)(BQ/QA) = 1.
Applying Menelaus's Theorem to ΔABC, cut by the line QR, we get
(AS/SC)(CR/RB)(BQ/QA) = 1.
Comparing these products AP/PC =  AS/SC.
From projective geometry, we know that AP/PC =  AS/SC, so either
of the theorems of Ceva and Menelaus implies the other. We call this
The CevaMenelaus Theorem
The theorems of Ceva and Menelaus are logically equivalent.
In fact, it is easy to show that Menelaus's Theorem implies Ceva's.
This was done, in hyperbolic geometry, to prove Ceva's Theorem.
The proof in euclidean (or affine) geometry is identical in structure.
I know of no way to prove the reverse implication except by using
projective geoemtry, or at least harmonic pencils.
