FINAL
ASSIGNMENT

Consider
triangle ABC. A point P was selected randomly inside the triangle.
segments were drawn from P to each of the triangles vertices and
then extended to intersect the sides of the triangle.

Let's
explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and
various locations of P.

First
I will explore for the triangle shown above.
Notice
that the products are equal.

Now
I will move the location of P.
The
product remains equal!

If
I make triangle ABC an obtuse triangle, the same relationship
holds.

Conjecture:
Can the results be generalized so that P can be OUTSIDE the triangle?
Notice,

here
I just extended the segments into dashed lines.

Notice
that when P is moved to the exterior of the triangle, the other
lines disappear.

Here,
I recreated the triangle constructing it with lines instead of
segments.

Now,
when I move the point P outside of the triangle, I can see that
the same relationship holds.
Click
HERE to move point
P around.

Next,
I will show that when P is inside triangle ABC, the ratio of the
areas of triangle ABC and triangle DEF is always greater then
or equal to 4.
Notice
that in this particular case, it is greater than 4.
Click
HERE to look at all
possibilities.

We
can see from the picture below that the ratios of the areas will
equal exactly four when the medial triangle is constructed.
RETURN