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**Final Assignment**

**By: Lauren Lee**

For this assignment, 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 in points D, E, and F respectively.

I have used GSP to construct triangle ABC with interior point P. Now I want to explore (AF)(BD)(EC) and (FB)(DC)(EA) for various
triangles and various locations of P.

For this ABC triangle example, I used GSP to find the measurements of
the sides. In this case:

Let’s look at more examples by moving the point P inside the triangle.

From these examples we can see that (AF)(BD)(EC) and (FB)(DC)(EA) are equal. Will they always be equal no matter where
the point P lies in the interior?

I want to make the conjecture that AF*BD*EC =
FB*DC*EA. In other words,

To prove this, I will use similar triangles. My first step is to construct parallel lines. I constructed lines through B and C that are
parallel to the line AD.

By using the alternate interior angle theorem and vertical angles, I
know that triangles DPC and BMC are similar and triangles BDP and BCN are
similar. I know from the properties of
similar triangles that:

__BD__ = __DP__ and __BC__ = __BM__

BC CN DC DP

By using the alternate interior angle theorem and vertical angles
again, I know that triangle BMF is similar to triangle APF and that triangle
CNE is similar to triangle APE. Now I
know from the properties of similar triangles:

__AF__ = __AP__ and __CE__ = __CN__

BF BM AE AP

Now I need to multiply
my equations together.

__BD*BC*AF*CE__ = __DP*BM*AP*CN__

BC*DC*BF*AE CN*DP*BM*AP

By simplifying I get:

__BD*AF*CE__ = 1

DC*BF*AE