| This is the write-up of the Final Assignment |
Brian R. Lawler
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| EMAT 6680 |
12/16/00
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| 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? |
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Again, first I constructed the initial conditions and measured the areas of the two triangles. At the right is a picture of the work. Click on the image to access a Geometer's Sketchpad file demonstrating the measurement. Allowing Geometer's Sketchpad to compute the area for several examples, it appears that 4 is the minimum ratio between the areas. It appears that this occurs when P is placed at the centroid of the triangle, the location where all the medians cross.
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| When P is placed at the centroid G, the result is four congruent triangles. Hence, the area of triangle DEF is 1/4 the area of triangle ABC. |  |
Next I returned to the notion of when P is placed anywhere
inside triangle ABC other than this centroid. Recall, it appears the ratio
of the areas is greater than 4. Suspecting I may gain some insight into
this from the conclusions in part B., I revisited this ratio. First,
(not really first, but this is where I am right now) I rewrote it as  .
And next, I constructed a visual representation of the areas represented
in the products on the right side of this equation. |
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Click on the image to see and investigate the sketchpad
file, which includes measurements and more of the construction lines as
well.
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This was significant because what I saw was that the rectangles formed
along the side BC became equal in height when P was at the centroid of
ABC. This seems to correspond to the fact that triangle AFE and triangle
BFD also become equal in height. When two triangles are equal in height,
the ratio of their areas is equal to the ratio of their bases. So I see
that |
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| I am stuck for where this may lead. And it is time to put away this problem. So I will leave this final conjecture unproved. | |
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Comments? Questions? e-mail me at blawler@coe.uga.edu |
| Last revised: January 3, 2001 |
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