Ceva

1. Ceva's theorem

Consider a triangle ABC and three points D on BC, E on CA and F on AB. Then a necessary and sufficient condition for these points that lines AD, BE and CF are concurrent at a point K or parallel (concure at a point at infinity) is
                                                             (DB/DC)(EC/EA)(FA/FB) = -1.
Here we take into account the orientation of the segments, so that (FA/FB) is negative for points F between A and B and positive for locations of F outside the segment AB. Thus, all factors of the above product are negative.

Three lines issued from the vertices of a triangle and concurring at a point K are called Cevians of the triangle. There are many famous cevians of a triangle, like the Medians, the Altitudes, the Bisectors, the Symmedians etc.

For the proof one can use Menelaus theorem (see Menelaus.html ) and apply it two times.
First for triangle ACF and the line EK, giving      (EA/EC)(BF/BA)(KC/KF)=1.
Then for triangle CFB and line KD giving         (DC/DB)(KF/KC)(AB/AF) = 1.
Multiplying both sides we get                               (EA/EC)(FB/FA)(DC/DB)=1.
This proves the necessity. The sufficiency is proved by contradiction and the use of the necessity. The argument is similar to the one applied in Menelaus.

Ceva's theorem is actually equivalent to Menelaus theorem. Here we proved that Menelaus implies Ceva. For the converse look at the file MenelausFromCeva.html .

Exercise prove the case where K tends to infinity, thus lines AD, BE and CF are all parallel to the same line.

2. Second version

Another form of Ceva's theorem we obtain by introducing two angles at each vertex defined through the respective cevian and the sides of the triangle. In the figure below the angles are such that A = A₂ - A₁, B = B₂ - B₁, C = C₂ - C₁. Then Ceva's condition is equivalent with ([Askwith, p. 41]).

This follows by observing that from the sinus theorem for triangles follows

Analogous equations to the last are valid also for the other vertices and multiplying these equations and simplifying we get at the stated condition.

3. Vectorial form

A third version of Ceva's condition is obtained by introducing unit vectors {u₁, u₂, u₃} respectively along the sides {AB,BC,CA} and {w₂,w₃, w₁} along the secants {AK,BK,CK}. Then, denoting by <...,...> the usual inner product and by J(X) the operator that turnsevery vector by π/2 we have the equivalent to Ceva's condition:

This follows from the second form of Ceva's condition by observing that the sinus of angles can be expressed by inner products:

Analogous formulas are valid also for the other angles and the condition follows by substitution into the condition of the previous section.

4. Projective form

A fourth version of Ceva's condition is obtained by intersecting the sides of the triangle and the cevians with an arbitrary line N as in the figure below. This defines on each side-line of the triangle one cross ratio (see CrossRatio0.html ) and the condition of concurrence is that the product of these cross ratios is -1 ([Green, p. 354]).
(BCEI)*(CAFK)*(ABDG) = -1.

From this results another condition which can be read out on line N from the six traces of the lines (sides + cevians), because of the relations
                                  (BCEI) =  (HJLI),       (CAFK) = (JLHK),     (ABDG) = (LHJG),
which imply
                                                            (HJLI)*(JLHK)*(LHJG) = -1.

The proof of the fourth version results from general principles of projective geometry, according to which for any two sets of four lines in general position there is a projective map transforming the first set onto the second. In particular, taking thefirst three lines to be the sides of the triangle, the fourth line of the first set to be N and the corresponding line in thesecond set to be the line at infinity we construct a map sending the cross ratios of the fourth version to ratios of points on the sides of the triangle, as these appear in the first version of Ceva's theorem. Thus the projective case isreduced to affine one handled in section-1.

5. Triangle + line

The statement on the existence of the projective map mapping a triangle + a line to another triangle + line, can be reduced to the well known property of projectivities to be completely determined by four points and their images. In fact, given the triangle and the line single out the four points {B,C,D,E} as in the figure below. Select also the corresponding four points {B',C',D',E'} in the secondsystem of triangle + line in the same way and define the projectivity by the obvious correspondence.

Bibliography

[Askwith] E. H. Askwith, A course of pure Geometry Cambridge University Press, Cambridge 1903
[Green] Green H.G. On the theorems of Ceva and Menlelaus American Math. Monthly, Vol. 64(1957)

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