Final Assignment

by

Shridevi kotta

Click here
to read the problem.

This write up explores and then proves the
required theorem. In this write up you will have geometrical illustrations to
follow along with the story. If you are interested in accessing the GSP file to
explore for various locations of a given point or for various shapes of the
triangle, click here to access the GSP file.

A..

We are given a triangle ABC and when point P is
inside the triangle. The ratios are as seen.

Moving the point to outside, we observe t he
following.

So, we observe by exploring for various
locations of P or for various shapes of triangle ABC that, the (AF)(BD)(EC) = (FB)(DC)(EA) or that their ratio is 1.

B. Now the proof goes as follows.

Hence we have,

Consider some more pairs of similar
triangles.

Hence

Thus proved the conjecture we made from
exploration in part A..

C.

We see that as long as P is contained inside the
triangle, DEF is bounded by the ABC and the ratio of ABC to DEF is 4 or greater
than 4.

Just as an observation, if P is outside the triangle
DEF can grow in size, and the ratio of triangle ABC to DEF can be smaller than
4.

We observe that the ratio is equal to 4 when P
coincides with centroid.