## EMAT 6680

## Final Assignment

#### By Kent Wiginton

I begin my final project by constructing a triangle ABC and
picking any point P inside the triangle. Next, construct the line
that passes through the vertex and p and constuct the intersecting
point on the opposite side.

We are posed with the following challenge:

### 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 inh points D, E, and F respectively.

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

###

From my observations, I can see that the products are the same.
(i.e. (AD)(BD)(EC) and (FB)(DC)(EA) are equal). So I set up an
equation to make sure that what I had observed was not a special
case. I found the ratio of the products and posted it on my GSP
window. And as I expected the ratio is equal to 1.

Another interesting observation is that when a vertex is moved
so that it lies on one of our constructed lines certain points
and sides become equal.

**See for yourself in GSP****.**
The shaded areas represent the range of movement of the vertex
where the other points still exist. Notice that when all three
triangles are the same we have a equalateral triangle, and P is
the orthocenter.

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