Quote:

Triangle $\displaystyle ABC$ is a 30-60-90 triangle, where $\displaystyle A = 30^o,\:B = 90^o,\:C = 60^o.$

Altitude BD is drawn to AC, forming $\displaystyle \Delta ADB$

Altitude DE is drawn to AB, forming $\displaystyle \Delta DE.$

Altitude EF is drawn to AC, forming $\displaystyle \Delta AFE.$

Altitude FG is drawn to AB, forming $\displaystyle \Delta AFG.$

Altitude GH is drawn to AC, forming $\displaystyle \Delta HG.$

Altitude HI is drawn to AB, forming $\displaystyle \Delta AHI.$

Find the ratio of the area of $\displaystyle \Delta AIH$ to the area of $\displaystyle \Delta ABC.$

I found that the resulting ratio $\displaystyle \frac{729\sqrt{3}}{8192}$

. . Close, but incorrect.