Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP and CP and extend them to intersect 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
Conclusion: The ratio (AF)(BD)(CE) : (BF)(CD)(AE) is always 1.

For a proof of this fact, please click here.

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