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Ratio of Triangles' Areas

Let \( AB = 4AA' \), \( BC = 4BB' \), and \( AC = 4CC' \). What is the ratio of the area of the inner triangle to the outer triangle?

Note: I have halfway solved this problem, in that I found the ratio empirically and found it to be a constant ratio independent of the particular triangle. I then found the ratio trigonometrically by letting $\triangle ABC$ be equiangular. However, I would much prefer to see a simpler proof that does not make any assumptions about the triangle itself. Here is a link to work I have already done on this problem.

Thanks!

Note by Andrew Edwards
4 years, 4 months ago

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It follows from Routh's Theorem that the ratio is 4:13.

Jon Haussmann - 4 years, 4 months ago

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Hint

Hint 2: Menelaus or Mass Points (my preference)

Daniel Chiu - 4 years, 4 months ago

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Thanks, I was able to prove it via Menelaus' Theorem.

Andrew Edwards - 4 years, 4 months ago

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