## Final Assignment

Part A: Exploration of various triangles and various locations of 'P':

Fromwhat we see, the ratio of AF*CE*BD/EA*DC*BF should be 1 in all cases. Why is that the case? Let's investigate:

Part B: Proof

Let's start out with P inside our traingle and draw parallel lines to all the sides through P.

 So we get ...

Next, let's investigate a few smaller sub-triangles.

From this, we get:

What we WANT are the sub-segment values (i.e. AF, DC, etc). Let's try to derive them from what we have:

Finally, some terms will start to cancel out. Let's look at individual segments first:

And finally, our conclusion:

Part C: Further Exploration

Let's examine the triangle formed by our P and its points (D, E, F):

It looks like our range for the ratio of areas is 4 --> infinity ... When will it be exactly 4?

To be exactly 4, w want F to be the midpoint of AB, E to be the midpoint of AC and D to be the midpoint of BC. Then we will have four congruent triangles (due to parallel lines and proportions of similiar triangles). This happens when P is the centroid.