Problem:
Equilateral triangle ABC has side length 840. Point D lies on the same side of line BC as A such that BD⊥BC. The line ℓ through D parallel to line BC intersects sides AB and AC at points E and F, respectively. Point G lies on ℓ such that F is between E and G,△AFG is isosceles, and the ratio of the area of △AFG to the area of △BED is 8:9. Find AF.
Solution:
Answer (336):
Let AF=FG=x. Then BE=840−x. Because △BED is a 30−60−90∘ triangle, BD=23(840−x) and DE=21(840−x). The altitude of △AEF, which equals the altitude of △AFG from A, is 2x3. Then
