Consider a triangle ABC and three points D on AC, E on BA and F on CB. Then a necessary and sufficient condition
for these points to be collinear on a line e is
(DA/DC)(EB/EA)(FC/FB) = 1.
Here we take into account the orientation of the segments, so that ratios of lengths of segments,like (FB/FA) are negative
for points F between A and B and positive for locations of F outside the segment AB.
A proof can be given by projecting segments AA*, BB* and CC* on line e and replacing the ratios entering the relation with the
ratios of the projecting segments, e.g. DA/DC = AA*/CC*, etc. ... The product becomes:
(DA/DC)(EB/EA)(FC/FB) = (A*A/C*C)(B*B/A*A)(C*C/B*B) = 1,
proving the necessity. The sufficiency can be proved by contradiction and use of the necessity. In fact, assuming the condition
and that line DE passes through another point F* of BC, different from F, one deduces that FC/FB=F*C/F*B, which implies the
1) For a proof of Menelaus on the basis of Ceva's theorem look at the file MenelausFromCeva.html .
2) This theorem implies (in fact, it is equivalent to) the theorem of Ceva, discussed in the file Ceva.html .
3) For a nice application of Menelaus to a property of parallelogramms look at the file MenelausApp.html .
4) An other application of the theorem can be found in Newton.html .
5) One of the most interesting applications of Menelaus is to the proof that the product of homotheties is again a homothety (or
translation in some cases), see the file Homotheties_Composition.html for a discussion on this.
6) The formulation used here for the ratios puts the dividing point at the beginning of segments e.g. DA/DC instead of using
AD/DA as is done in other references e.g. [Bottema, p. 85], where the product is then -1 instead of +1.
The condition of Menelaus theorem can be generalized to cross ratios (see CrossRatio0.html ) by introducing an arbitrary line L,
which together with the other four lines builds a set in general position i.e. no three lines of the set pass through the same
point. The figure below shows such a figure. The condition of Menelaus is then equivalent to ([Green, p. 354])
(ABDJ)*(BCFI)*(CAEG) = 1.
The proof can be given as in section-4 in the case of Ceva's theorem (see Ceva.html ). Define the projective transformation which
maps the side-lines of the triangle, each to itself and the line L to the line at infinity. Then the cross ratios are preserved and
transfer to ratios in the case of line at infinity. Then apply the Menelaus theorem in its usual form.
[Bottema] O. Bottema Topics in Elementary Geometry Springer Verlag Heidelberg 2007
[Green] Green H.G. On the theorems of Ceva and Menlelaus American Math. Monthly, Vol. 64(1957)
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