Triangle is a 30-60-90 triangle, where
Altitude BD is drawn to AC, forming
Altitude DE is drawn to AB, forming
Altitude EF is drawn to AC, forming
Altitude FG is drawn to AB, forming
Altitude GH is drawn to AC, forming
Altitude HI is drawn to AB, forming
Find the ratio of the area of to the area of
I found that the resulting ratio
. . Close, but incorrect.
Note that all triangles are 30-60-90 right triangles.
In a 30-60-90 triangle, the side opposite 30° is half the hypotenuse.
The side opposite 60° is times the shortest side.
Let , then
The area of is: . .
. . The area of
We have: .
Drawing two altitudes, we have two similar right triangles.
. . The area of the smaller triangle is of the larger.
This procedure is performed three times.
Hence, the area of the final triangle is: . the area of