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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