Theorems of Menelaus, Pascal and Ceva: five solved problems proposed for the International Mathematics Olympiad

Abstract

The Theorems of Menelaus, Pascal and Ceva establish necessary and sufficient conditions for the concurrence and collinearity of certain lines and points. Five problems proposed for the International Mathematics Olympiad are discussed in detail. The proofs involved in the solutions are complemented by the availability of the respective links to the interactive figures using GeoGebra. It is expected that the article can be appreciated both by students preparing for the final stages of national or international competitions and by teachers interested in Olympic challenges. Other topics that appear are: quadrilaterals, similarity, point power, median base, incenter and homothety.