Waste less time on Facebook — follow Brilliant.
×

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
3 years, 11 months ago

No vote yet
5 votes

Comments

Sort by:

Top Newest

It follows from Routh's Theorem that the ratio is 4:13. Jon Haussmann · 3 years, 11 months ago

Log in to reply

Hint

Hint 2: Menelaus or Mass Points (my preference) Daniel Chiu · 3 years, 11 months ago

Log in to reply

@Daniel Chiu Thanks, I was able to prove it via Menelaus' Theorem. Andrew Edwards · 3 years, 11 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...