Perpendicular Trisectors and Base Trisectors in Triangles and Quadrilaterals

The four centers of a triangle- namely centroid, circumcenter, orthocenter, and incenter are well known. These are characterized by some famous lines known as median, perpendicular bisector, altitude, and angle bisector. Bisectors, whether side bisector or angle bisector, are commonly seen. Trisectors, both angle trisectors and base trisectors, also have interesting properties, including Morley’s theorem (property of angle trisectors). The properties of other trisectors have been overlooked and this paper aims to highlight them. The research objective of this paper is to review the properties of base trisectors, perpendicular trisectors and n-section in Quadrilateral. We explore the properties and proofs and finally get a generalization to the quadrilateral trisection theorem. The major findings of this research work are the discovery of similar triangles in trisectors, leading us to propose i) base trisectors theorem, ii) perpendicular trisectors theorem, and iii) quadrilateral n-section theorem.