We will begin with a verification of Ceva's Theorem.
Given triangle ABC with cevians BG, AF and CE concurrent at point D, we wish to demonstrate that:
Click here for a GSP 4.0 sketch to investigate.
We begin by constructing altitudes h1 and h2. The area of triangle ABG [in the future, we will use the notation (ABG) to designate the area of triangle ABG] equals 0.5(AG)(h1) and (BGC) = 0.5(GC)h1. Also, (ADG) = 0.5(AG)(h2) and (DGC) = 0.5(GC)(h2). Thus,
We conclude, then, that
Similar reasoning yields the equations
We will now consider the converse of Ceva's theorem.
show that cevians BG, AF and CE are concurrent.
Click here for a GSP 4.0 sketch to manipulate.
Assume that cevians AF and CE intersect at D, and that the other cevian through D is BH. By Ceva's theorem,
Since we assumed
by the transitive property,
which is true only if H and G represent the same point. Thus, segments AF, CE and BG must be concurrent.
(Reference: Coxeter, H. S. M. & Greitzer, S. L. (1967). Geometry Revisited. Washington D. C.: The Mathematical Association of America. 4.)