Final Assignment

Emat 6680 with Dr. J. Wilson

Page Bird

Complete a Write-up on your Web Page for the following investigation. This should be individual work.

A. Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D, E, and F respectively.

Explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.

Below are some of my explorations.

When P is inside the triangle:

When P is outside the triangle:

When P lies on a side of triangle ABC.

The GSP sketches here are pretty convincing that .

But the question is how do we prove this. Which brings us to part B of this assignment.

B. Conjecture? Prove it! (you may need draw some parallel lines to produce some similar triangles) Can the result be generalized (using lines rather than segments to construct ABC) so that point P can be outside the triangle? Show a working GSP sketch.
Using the parallel lines below, let's look at a proof.

By similar triangles and .

We can also make the relationship, and by vertical angles.

When we multiply these relationships, we get .

C. Show that when P is inside triangle ABC, the ratio of the areas of triangle ABC and triangle DEF is always greater than or equal to 4. When is it equal to 4?
Here is one example I looked at. In each example, the ratio was always greater than or equal to 4. I found one example when the ratio was equal to four. Thei occurs when triangle DEF has its vertices at the midpoints of the sides of the triangle. Click here to see a GSP dynamic sketch.