The central topic of this thesis is a theorem that is usually attributed to the outstanding English mathematician E. J. Routh, since he was apparently the first one to publish that particular statement. It was included as a helpful note, without a proof, in his book on statics. If the vertices of a triangle \(ABC\) are joined to three points chosen on the opposite sides, another triangle \(PQR\) is formed by mutual intersection points of these segments. Routh gave a formula for the ratio of the areas of \(PQR\) and \(ABC\), which can be easily transformed into a function of the ratios \(r, s\) and \(t\), determined by the points chosen on the sides. The possibility of different approaches by various methods, as well as its obvious connection to the classical theorems of Menelaus and Ceva, prompted numerous mathematicians to find new proofs and generalizations of Routh’s formula. Some of these results are presented here, including a unification of Menelaus’ and Ceva’s theorem by introducing a so-called generalized Routh’s triangle. The final chapter is dedicated to a three-dimensional analogue of Routh’s theorem, valid for any tetrahedron and entirely corresponding to the basic formula for a triangle.