## Exploring the products: (AF)(BD)(CE) and (AE)(BF)(CD)

#### by: BJ Jackson

In this section, we look at what happens to the products as
the point p changes positions in the interior of the triangle
and as the lengths of the sides of the triangle change.

We start by making our diagram and finding the products. GSP
will do all of this for us and we find our intitial values for
the picture below to be:

and .

The products are the same! Now, we have to wonder if this is
always true or is this just a random happening. To try and figure
this out, we will vary the location of point P and the lengths
of the sides. If this product continues to be the same, then we
will be able to make a conjecture that the products are always
the same no matter where the point p is located in any triangle.

So, we will try some different locations for P and find the
products.

The products remain constant. Click **here**
for a GSP sketch that will animate the point p inside the triangle.
One will see that no matter where the point goes, the product
stays the same.

Now, we will vary the size and the shape of the triangle and
check the products.

Again, the products remain the same. If one wants change the
sides of the triangle, click on the link above for the animation
and you can vary the shape and size of the triangle.

This leads us to believe that our conjecture that the products
are always equal is true. Another way of stating the conjecture
is to say that the ratios of the products equals 1 which is what
will actually be proven.

**A proof of the conjecture**

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