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.