Part A: 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.
We'll examine a few different triangles with different side measures and change the location of P for each. For each of these we will compare the products of (AF)(BD)(EC) and (FB)(DC)(EA) and see of there is any relationship:
EXAMPLE 1:
EXAMPLE 2:
EXAMPLE 3:
Part B: Conjecture? Prove it!
Because (AF)(BD)(EC) and (FB)(DC)(EA) are always equal, I make the conjecture that the ratio of these to products is equal to 1.
In order to prove this conjecture, I will need to use similar triangles.
Proof: Make two lines that are parallel to segment BD through points A and C.
Because the lines are parallel, I can use the Alternate Interior Angles Theorem and the Vertical Angles Theorem to get similar triangles.
I know that triangles EPC and AGC are similar and triangles AEP and ACH are similar.
Because I have similar triangles, I can set up the following ratios:
and
Triangles AGF and BPF and triangles CHD and BPD are also similar.
Because I have similar triangles, I can set up the following ratios:
and 
Therefore, using a property of proportionality:
Simplify the expression and the proof is complete:
