Triangle ABC is a 30-60-90 triangle, where A is 30º, B is 90º, and C is 60º. A segment BD is the drawn altitude to AC, forming ΔADB, then DE is the altitude drawn to AB,forming new ∆ADE. Then, EF is the altitude drawn to AC, forming new ∆AFE. 78Then, FG is the altitude drawn to AB, forming new ∆AFG.Then, GH is the altitude drawn to AC, forming new ∆AHG. Then, HI is the altitude drawn to AB, forming new ∆AHI. Find the ratio of the area of ∆AIH to the area of ∆ABC?

I found that the resulting ratio 729√(3)/8192 : √3/2 or .154 : .866