## Part B: Proof that the Ratio of the Products = 1

#### by: BJ Jackson

This proof hinges on the diagram that was used in Part A and the addition of a line that is parallel to side BC through the point A. This proof would work the same in any of three ways as long as the parallel line goes through one of the vertices of the triangle and is parallel to the side opposite of the vertex. For this proof, the following diagram will be used.

Here, we want to prove that . This will be done by using similar triangles and the ratios of the sides of similar triangles.

There are now several similar triangles that can be found using vertical angles and parallel lines cut by a transversal. These similar triangles yeild several useful proportions. The proportions are:

, , , and .

Next, we multiply the proportions and reduce the fractions and we get

.

Which is what we wanted to prove. So, the only way for the proportion to equal 1 is if their products are equal. So this proves our conjecture that