Os teoremas de Menelaus e Ceva

Abstract

The present work presents two important theorems of plane geometry. The first deals with the collinearity of three points on the straight brackets of the sides of a triangle, known as the theorem of Menelaus, and dates from the 1st century. The second is the Ceva’s theorem, dating from the 17th century and refers to the competition of three segments connecting each vertex to any point on the opposite side of a triangle. We present different demonstrations of these theorems, using known concepts of geometry, such as: proportionality of segments, congruence and similarity of triangles, area calculation, trigonometry, vector geometry and barycentric coordinates. We treat the corresponding versions of such theorems in the space geometry. Initially, we provide historical data of the theorems and conclude with some applications.