## 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.

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**.

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