**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.**

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**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:**

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**EXAMPLE 1:**

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**EXAMPLE 2:**

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**EXAMPLE 3:**

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**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.**

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**In order to prove this conjecture, I will
need to use similar triangles.**

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**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.**

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**I know that triangles EPC and AGC are similar
and triangles AEP and ACH are similar.**

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**Because I have similar triangles, I can
set up the following ratios:**

**Triangles AGF and BPF and triangles CHD
and BPD are also similar.**

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**Because I have similar triangles, I can
set up the following ratios:**

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**Therefore, using a property of proportionality:
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**Simplify the expression and the proof is
complete:**