4.6 Concurrency Theorems

. Concurrency Theorems Definition A cevian is a line segment that extends from one vertex of a triangle to the opposite side. In the diagram, AD is a cevian, from A . Special cevians (i) A median is a cevian that divides the opposite side into two congruent(equal) lengths. (ii) An altitude is a cevian that is perpendicular to the opposite side. (iii) An angle bisector is a cevian that bisects the corresponding angle. Ceva’s Theorem (without proof) Statement Let ABC be a triangle and let D,E,F be points on lines BC , CA, AB respectively. Then the cevians AD, BE, CF are concurrent if and only if BD DC CE EA AF FB = where the lengths are directed. This also works for the reciprocal of each of the ratios as the reciprocal of is . Giovanni Ceva (Dec , – June , ) In , Ceva was designated as the professor of Mathematics, University of Mantua and worked there for the rest of the life. In , he published an important theorem on synthetic geometry for a triangle called Ceva’s theorem. Ceva also rediscovered and published in the Journal Opuscula mathematica and Geometria motus in . He applied these ideas in mechanics and hydraulics.