Ceva's theorem

The set of Ceva is a geometric statement about Dreieckstransversalen, the ( 1647-1734 ) in 1678 showed the Italian mathematician Giovanni Ceva in his work De lineis rectis.

In a triangle ABC are [AD ], [ BE] and [ CF] three vertex transversals (ie links between a corner and a point on the opposite side or its extension ), which intersect at a point O inside or outside the triangle. Then:

Here is the ( oriented, so possibly negative ) part ratio, which for three points lying on a straight line with defined is. If is between and, the said division ratio is the same, otherwise the same.

The above equation can be proved using the theorem of Menelaus.

Conversely, it can be inferred from the correctness of this equation that intersect the lines AD, BE and CF at one point. This reversal of the sentence of Ceva is often used in the triangular geometry for evidence from the field " Excellent points in the triangle ."

If the equation is true, it also follows that:

Since the orientation is lost in this case, this equation is not sufficient for a reversal of the sentence, see Theorem of Menelaus.

A generalization of Ceva is the set of Routh.