Routh's theorem

The Ceva's theorem is one of the most important theorems in elementary geometry. This theorem provides criteria under which a set of three Ceva's line segments, one through each vertex and a point of opposite lying side of the given triangle are concurrent. The Routh's theorem is a kind of generalization of the Ceva's theorem. When the given Ceva's lines are not concurrent, the Routh's theorem gives the ratio between the areas of the given triangle and the triangle, which we get with the intersection of the Ceva's lines. In this work we present and prove the Routh's theorem with the help of the Menelauses' theorem. In the last part of this work we present the generalization of the Routh's theorem to the case when six Ceva's line segments are given, one pair through each vertex of the given triangle