Having made the conjecture that the ratio of (AF)(BD)(CE) to (BF)(CD)(AE) is always equal to 1 whether P is located in the interior or exterior of triangle ABC, it follows that some proof is required.
To begin, construct lines parallel to segment FD through B, to segment FE through A, and to segment DE through C. Follow this construction by constructing the points of intersection of these parallel lines (O, Q, and R). Next, construct line segments OF, DQ, and RE as shown.
We are interested in triangles BOA, QBC, ARC; specifically the ratio of the areas of the red triangles to the blue triangles as shown.
As shown in the above illustration, the ratio of interest is equal to 1, just as the primary ratio being investigated is equal to one. This relationship holds for any location of P. Click here to try.
It is also interesting to compare the ratio of areas of triangles ODR and ABC to the ratio of the areas of triangles ABC and FED. As seen in the following illustration, they are equal. This is true for all locations of P. Click here to explore.
