To start this proof, let's recall a theorem from geometry called Menelaus' theorem. First, we need to know that Menelaus points are simply points, which are on the lines containing the sides of a triangle. The theorem says that if there is a triangle ABC with collinear Menelaus points X, Y, and Z on the lines extended from the sides AC, BC, and AB, respectively, then:
The product of the ratios of these segments is equal to -1 because the measurements of the segments were directional.
We can now use this information in the proof of the conjecture we made earlier. In our triangle ABC, we can take triangle CFB within this. On that triangle, we have collinear Menelaus points A, P, and D after BF is extended.
We have applied Menelaus' theorem to show thatThe ratio of FA to AB will be negative because their directions are opposite of each other. We could choose either segment as our negative direction, and we still get -1.
Next, we can choose triangle ACF and use Menelaus' theorem like before to show that
Now, let's multiply the left and right sides of these two equations together:
The right side is equal to 1 because it is -1 times -1. Now, we can simplify this equation and finally get:
The terms here can be rearranged to prove our original conjecture that (AF)(BD)(CE) = (BF)(CD)(AE).