# FINAL ASSIGNMENT

Part 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 in points D, E, and F respectively.

Consider the following triangle:

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

We'll examine a few different triangles with different side measures and change the location of P for each.  For each of these we will compare the products of (AF)(BD)(EC) and (FB)(DC)(EA) and see of there is any relationship:

EXAMPLE 1:

EXAMPLE 2:

EXAMPLE 3:

From these examples, we can see that no matter the triangle nor the position of P inside the triangle, (AF)(BD)(EC) and (FB)(DC)(EA) are always equal.

Part B:  Conjecture?  Prove it!

Because (AF)(BD)(EC) and (FB)(DC)(EA) are always equal, I make the conjecture that the ratio of these to products is equal to 1.

In order to prove this conjecture, I will need to use similar triangles.

Proof: Make two lines that are parallel to segment BD through points A and C.

Because the lines are parallel, I can use the Alternate Interior Angles Theorem and the Vertical Angles Theorem to get similar triangles.

I know that triangles EPC and AGC are similar and triangles AEP and ACH are similar.

Because I have similar triangles, I can set up the following ratios:

and

Triangles AGF and BPF and triangles CHD and BPD are also similar.

Because I have similar triangles, I can set up the following ratios:

and

Therefore, using a property of proportionality:

Simplify the expression and the proof is complete: