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 respectively.

Figure 1

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
seems that 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, , , , & .

Figure 3

Therefore we have , , , & respectivly.

From this we get

then

then we finally get

.

We use this along with & , and substitute into .

So we get , where . 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.

Figure 4

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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