**Final Project**
**Nicole Mosteller**
**EMAT 6680**

Given point **P** inside **DABC** with segments **AD**, **BE**, and **CF** containing point **P**.
Investigate the relationship between **(AF)(BD)(CE)** and **(FB)(DC)(EA)**.

From the investigation of **DABC**, the following data was collected:
Similar data was collected for various locations of point P.
This investigation has led me to the following conjecture:
Given point **P** inside **DABC** with segments **AD**, **BE**, and **CF** containing point **P**,
then the ratio **(AF)(BD)(CE)**:**(FB)(DC)(EA)** is **1**.

To begin this proof, it is necessary to add a few auxillary lines to **DABC**. The auxillary lines that have been added
are parallel to **BE** - one through point **A** and the other through point **C** (see **Figure 1**).
__Figure 1:__ Additional parallel lines through **A** and **C**.

In addition to the parallel lines, I have altered the original **DABC** by extending **CP** to intersect the new line through **A**
at point **J** and by extending **AP** to intersect the new line through **C** at point **K** (see **Figure 2**).
__Figure 2:__ Extended lines.
These extended lines in addition to the parallel lines help to create multiple similar triangles.
**Figure 3**, **Figure 4**, and **Figure 5** show the data that comes from the multiple similar triangles.

__Figure 3:__ **DAJC** is similar to **DEPC** (by **AA**).
.
From this set of ratios, we see that
.

__Figure 4:__ **DAPE** is similar to **DAKC** (by **AA**).
.
From this set of ratios, we see that
.

__Figure 5:__
**DAFJ** is similar to **DBFP **(by **AA**). **DPDB** is similar to **DKDC **(by **AA**).
Using the multiplication property, we see that
.
This simplifies to
.
Substituting our data from **Figure 3** and **Figure 4**, we see that
.
This simplifies to
and allows us to conclude that
.

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