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.

 

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:

 

 

 

Letís 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:

 

 

EXAMPLE 1:

EXAMPLE 2:

EXAMPLE 3:

*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!

 

 

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.

 

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

 

I will need to construct parallel lines: Make two lines that are parallel to segment AD through points B and C.

 

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.

 

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

 

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

 

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.

 

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

 

I can now multiply these equations together:

 

Though algebra and simplifying, the result is:

 

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?

 

 

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.

 

Now, letís 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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