Complementary Halfspaces and Trigonometric Ceva–brocard Inequalities for Polygons

The product of ratios that equals 1 in Ceva’s Theorem is analyzed in the case of non-concurrent Cevians, for triangles as well as arbitrary convex polygons. A general lemma on complementary systems of inequalities is proved, and used to classify the possible cases of non-concurrent Cevians. In the concurrent case, particular consideration is given to the Brocard configuration defined by equal angles between Cevians and polygon sides.