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Final Assignment**

**by Laura Evans, UGA**

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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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I started with the following
triangle:**

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***Explore
(AF)(BD)(EC) and (FB)(DC)(EA) for various triangle and various locations of P:**

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Let’s look at 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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*NOTICE: From these examples we can see that no mater the triangle or the
position of P inside the triangle, (AF)(BD)(EC) and (FB)(DC)(EA) are always
equal.**

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**Part B:
Conjecture? Prove it!
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Because I have noticed that (AF)(BD)(EC)
and (FB)(DC)(EA) are always equal, I would like to 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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I will need to construct
parallel lines: Make two lines that are parallel to segment AD through points B
and C.**

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Because I have constructed
parallel lines, I can now use the theorem that alternate interior angles of
parallel lines are congruent and the theorem that says that vertical angles are
congruent.**

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Now I know that triangles DPC
and BGC are similar and triangles BDP and BCH are similar.**

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By the properties of similar
triangles I can set up the following ratios:**

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Using the properties of
alternate interior angles and vertical angles, I found that triangles BGF and
APF are similar and triangles CHE and APE are similar.**

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By the properties of similar
triangles I can set up the following ratios:**

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I can now multiply these
equations together:**

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Though algebra and simplifying,
the result is:**

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PROOF COMPLETE!!!**

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

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First I will start with triangle
ABC with a random point P inside the triangle:**

*Notice: The ratio of the areas is greater than 4 when a random point P is chosen inside the triangle.

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Now, let’s find the centroid of
the of triangle ABC, constructing triangle DEF inside triangle ABC:**

*Notice: When point P is the centroid of the triangle, the ratio of the areas of triangle ABC and DEF is 4.

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