## Part C: The Ratios of the Two Triangles

#### by: BJ Jackson

For this part of the final, we will use the following diagram

to show that the ratio of triangle ABC to triangle DEF is greater
than or equal to 4. Again, we use GSP to make all of our calculations
and find the initial ratio which is 4.26. So, we get a value that
was expected.

Now, we can move point p anywhere on the inside of the triangle
and see that the ratio is always greater than or equal to 4. Then,
we change the size of the triangle by moving the vertices and
the ratio is still greater than or equal to 4. Click **here**
to open a GSP sketch that will allow you see this for yourself.

The truly interesting aspect of this is finding when the ratio
is exactly 4. So, I proceeded to use a script from assignment
5 that creates the triangle centers. They can be seen on the diagram
above labeled H, I, G, and C. Once we have the centers, it is
easy to drag point p on top of the centers to see if one of them
gives the desired ratio.

I started having P = H and found the following

which is not 4. So, then I tried P = I

which was very close but not equal to 4. Next, I checked P
= C

which was worse. So then, that leads one to believe that the
ratio of 4 is between G and C, so finally I let P = G and found

which is exactly what we are looking for. Therefore, the ratio
of the two triangles equals 4 when the point on the interior of
triangle ABC is the centroid.

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