Final Assignment: Project
Problem: 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
Explore (AF)(BD)(CE) and (AE)(BF)(CD) for various triangles and various
locations of P.
After moving point P to various
locations around the inside and outside of different triangles it
for allcases. Click Figure 2 below to try
moving point P yourself.
Now I want to prove that .
The segments BP, AP, & CP were extended and line HG was constructed
parallel to side AB and through point C. See figure 3 below. There are
several similar triangles, , , , & .
Therefore we have
From this we get
we finally get
We use this along with
, and substitute into
So we get
. So this proves
Let's now take a look of the ratio of the area of
triangle ABC to triangle DEF when point P is inside of triangle ABC.
Using GSP and moving point to various locations within triangle ABC,
and changing the shape of triangle ABC the ratio of the area of
triangle ABC to triangle DEF when point P is inside of triangle ABC was
always above 4.0. Click on the figure below to try for yourself.
When the point P coincides with the centroid G of the triangle ABC the
ratio of the areas is a minimum of 4.00. This makes sense because when
point P coincides with the centroid C the segments DE, EF, & FD
form the medial triangle for triangle ABC. The area of the medial
triangle has 1/4 the area of triangle ABC, therefore the ratio at this
point will be 4, and as point P moves away from the centroid G the area
of triangle DEF gets smaller so the ratoi will increase.
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