Philippa M. Rhodes

# Final Project

## Part I., #3

Consider any triangle ABC. Select an arbitrary point P and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D, E, and F respectively.

Explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.

(AF)(BD)(EC) = 2.27 cubic inches and (FB)(DC)(EA) = 2.27 cubic inches. Thus,

We will use similar triangles to prove this. To construct the similar triangles, draw two lines parallel to segment FC. One through point B, and the other through point A.

Notice that there are several pairs of similar triangles.

Recall

1. If two parallel lines are cut by a transversal (a line that intersects two or more lines in the same plane at distinct points), then the alternate interior angles are congruent, the alternate exterior angles are congruent, and the corresponding angles are congruent.

2. Vertical angles are congruent.

3. AA ~ AA. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

4. Corresponding sides of similar triangles are proportional segments.

Here, ARB ~APF.

so,

Next, AGB ~ FPB

so,
.

Also, AGE ~ CPE.

Thus,
.

The last pair of similar triangles that is used for this proof are CDP and BDR.

Hence,
.

Now we can use the previous ratios to reduce

We see that

and .

Thus,

.

Next, just subtitute.
and to obtain

###### . QED

Click here (for GSP file) to move P or here to change the triangle.