Choose 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.
Click here to explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.
Click here to see a working GSP sketch.
When P is insides triangle ABC, the ratio of the areas of triangle ABC and triangle DEF is always greater than or equal to 4.
Click here to see a GSP animation.
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