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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
.