**Problem (Parts A and B): **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. Explore(AF)(BD)(ED)
and (FB)(DC)(EA) for various triangles and various locations of
P.

**Click here for
a GSP sketch to investigate:**

**Conjecture:**

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for a proof of this conjecture:**

**Problem (Part C): Show
that when P is inside triangle ABC, the ratio of the areas of
triangle ABC and triangle DEF is always greater than or equal
to 4. When is it equal to 4?**

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a GSP sketch to investigate:**

**Conjecture #1: The ratio of triangles
ABC and DEF equals 4 when P is located at the centroid.**
**Click here
for a justification of the conjecture:**