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!

#Geometry #MathProblem #Math

Easy Math Editor

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Comments

Hint

Hint 2: Menelaus or Mass Points (my preference)

Daniel Chiu - 7 years, 9 months ago

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

Andrew Edwards - 7 years, 9 months ago

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

Jon Haussmann - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$