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.