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December 6th, 2010, 08:03 PM   #1
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triangle geometry

O is any point inside the triangle ABC. AO, BO, CO are joined and produced to meet the opposite sides BC, CA, and AB at D, E and F respectively. Prove that AO/AD+BO/BE+CO/CF=2

Obviously, O is meant to be the centroid. The question did not say that the lines produced cut the opposite sides into two equal lengths, where D, E and F are midpoints of the respective sides they cut. Shouldn't the question include this piece of information?
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December 6th, 2010, 08:36 PM   #2
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Re: triangle geometry

Why does it necessarily have to be the centroid?
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December 7th, 2010, 08:41 AM   #3
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Re: triangle geometry

Quote:
Originally Posted by jcmc2112
Why does it necessarily have to be the centroid?
Isn't it? Any suggestion to proving this?
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December 7th, 2010, 02:51 PM   #4
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The lines AO, BO and CO divide triangle ABC into various triangles, whose areas are related to each other in certain fairly obvious ways. The equation you have to prove can be written as a relationship between those areas, and can be shown to follow from the relationships I mentioned previously.
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December 8th, 2010, 07:46 AM   #5
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Re:

Quote:
Originally Posted by skipjack
The lines AO, BO and CO divide triangle ABC into various triangles, whose areas are related to each other in certain fairly obvious ways. The equation you have to prove can be written as a relationship between those areas, and can be shown to follow from the relationships I mentioned previously.
Thanks Skipjack. I made a sketch and there are quite a number of triangles here, which of them are involved?
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December 8th, 2010, 11:29 AM   #6
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Most of them. You might be able to avoid explicitly mentioning a very few. Various pairs of triangles have an altitude in common, making it worthwhile to consider the ratio of their areas. You can get a good idea of the type of proof you need to find by looking up a proof of Ceva's theorem, and you'll need to come within a whisker of proving that theorem as part of solving this problem.
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December 8th, 2010, 08:37 PM   #7
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Re:

Quote:
Originally Posted by skipjack
Most of them. You might be able to avoid explicitly mentioning a very few. Various pairs of triangles have an altitude in common, making it worthwhile to consider the ratio of their areas. You can get a good idea of the type of proof you need to find by looking up a proof of Ceva's theorem, and you'll need to come within a whisker of proving that theorem as part of solving this problem.
Thank you, that's really helpful.
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