Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D, E, and F respectively. Explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.

I began by constructing the initial conditions. At the right is a picture of the work. (Throughout the document, click on images to access an associated Geometer's Sketchpad file.)

By measuring each of the lengths above and computing the products, it is apparent that these products are always equal.