My previous column suffers from a conspicuous omission. I have quoted Dan Pedoe,

but then went on and discussed only a fine proof of Ceva's theorem. Menelaus' theorem was never mentioned again. This time, I'd like to give the latter its due. Here are the two theorems side by side:

Ceva's Theorem

Three Cevians AD, BE, and CF are concurrent if and only if

holds.

Menelaus Theorem

Let three points F, D, and E, lie respectively on the sides AB, BC, and AC of ABC. Then the points are collinear if snf only if

At first sight, (1) and (2) express exactly the same fact: that the product of three ratios of segments on the three sides of the triangle equals 1. Now, if (1) and (2) are the same, how can they be equivalent to the two essentially different facts? And the facts enunciated by the theorems of Ceva and Menelaus are indeed different. Therein lies a question, but also a clue to an answer.

Both theorems allow the points D, E, and F to lie not only on the sides of ABC but also on the extensions of the sides. In fact, the theorem of Menelaus requires that at least one of the points lie on the extension of the corresponding side due to an obvious fact that a straight line can't cross all three sides of a triangle internally. (As an aside, in a 1945 Russian mathematics competition, one boy did not see that fact as obvious. For that insight, he was awarded the first prize although he did not solve a single problem.)

Of the three points D, E, or F, none, one, two or all three may lie externally to the triangle, on the side extensions. If 1 or 3 points lie on extensions, we have the Menelaus theorem. In the other two cases, the theorem is Ceva's. It then appears that the formulations of the theorems suggest a question. For example, Menelaus' should have read

Let three points F, D, and E, lie respectively on the sides AB, BC, and AC of ABC or their extensions. Assume that either just one or all three of the points lie on side extensions. Then the points are collinear if and only if (2) holds.

Although cumbersome, the latter formulation must be preferred, unless of course there is a better way to rectify the situation. The most common approach avoids ambiguity by considering signed segments. By convention, for any two points P and Q, PQ and QP denote segments of different signs such that PQ = - QP. (It is also possible to assign a direction on the line PQ thus endowing each of the segments with a proper sign. Regardless of the choice of direction, we shall have PQ = - QP, the only important relation for the present.) In general, two segments (on one and the same line) looking into the same direction have the same sign, those looking into opposite directions have different signs.

For three points A, F, B, the ratio AF/FB is positive if F lies between A and B, and is negative if F lies on extension of the segment AB. It is also always true that AF/FB = -AF/BF. For Ceva's theorem, either all three points D, E, and F lie on the corresponding segments BC, CA, and AB internally, or some two of them lie on segment extensions, i.e. externally. Accordingly, in (1), either all three ratios are positive or exactly two of them are negative. In both cases the product is 1. For the Menelaus theorem, the product in (1) is negative, while the product in (2) is negative for Ceva's theorem.

If we are bent on using the same product for both theorems, then there are two possibilities:

The lines AD, BE, and CF in Ceva's theorem are called Cevians. The theorem gives a necessary and sufficient condition for the concurrency of three Cevians. A straight line is often called a transversal to emphasize its relation to another shape. The Menelaus theorem gives a necessary and sufficient condition for three points -- one on each side of a triangle -- to lie on a transversal. What is a Cevian in one triangle is a transversal in another. For example, the Cevian BE serves as a transversal in ADC while CF is a transversal in ADB. Write condition (2) for the two triangles:

Eliminate AK/DK from the two identities and recollect the sign convention (CB = -BC, etc.) The effort should lead to (1) which shows that the Menelaus theorem implies that of Ceva. (More accurately, this shows that (1) holds for three intersecting Cevians. The converse follows in a standard manner by the reductio ad absurdum argument. The necessary and sufficient conditions in the Menelaus theorem also follow from each other in a similar manner.) The theorems are indeed equivalent. But the proof deserves a page of its own.

There are great many proofs of the theorem of Menelaus. I'll give just two. One is the most economical in terms of required constructions (just one additional line), the other highlights an unexpected link between the theorem and other geometrical concepts.

Proof #1

Draw AP parallel to DE. Triangles ABP and BDF are similar as are triangles ACP and CDE. The first pair gives AF/BF = PD/BD, the second AE/CE = PD/CD. Combine the two to get (2).

Proof #2

From the vertices of ABC drop perpendiculars on the transversal. Consider three pairs of similar triangles AH_aF and BH_bF, CH_cD and BH_bD, and AH_aE and CH_cE. From these we get

Multiply the three to obtain (2).

The latter proof is suggestive. Points D, E, and F serve as centers of homothety for pairs of similar shapes - segments in this case - BH_b and CH_c, AH_a and CH_c, and AH_a and BH_b. The Menelaus theorem then says that, given three shapes two of which were obtained from the third by central similarity transformations, then (naturally) there exists a homothety that transforms the first into the second, and centers of all three transformations are collinear.

As a bonus, we obtain a solution to the following problem of three circles:

Furthermore, from the foregoing discussion it follows that two pairs of tangents may be taken internally.

The simplicity of the Menelaus theorem is deceptive. A few applications of (2) yield easily theorems of Desargues, Pappus, and Pascal. Another striking application can be found in [Honsberger].

At each vertex of a triangle there is a couple of angle bisectors: a bisector of the interior angle and a bisector of the exterior angle. It's well known that the bisectors taken one for each vertex are concurrent provided all or two of the bisectors are internal. This is a particular case of Ceva's theorem. In general, every angle bisector crosses the opposite side of the triangle. It then follows from the Menelaus theorem, that every three such points are collinear provided all or two of the bisectors are external.

References

1. H.S.M.Coxeter, S.L.Greitzer, Geometry Revisited, MAA, 1967
2. R.Honsberger, Episodes in Nineteenth and Twentieth Century Euclidian Geometry, MAA, 1995
3. D.Pedoe, Geometry: A Comprehensive Course, Dover, 1970

Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at alexb@cut-the-knot.com

Copyright ® 1997-1999 Alexander Bogomolny