# Consecutive Vertices of a Quadrilateral

$$D, G$$ are points on the side $$AB$$ of $$\triangle ABC$$. $$E$$ and $$F$$ are points on the sides AC and BC respectively such that $$DE \parallel BC,$$ $$EF \parallel AB$$ and $$FG \parallel CA$$. Then $$D, E, F, G$$ are the consecutive vertices of a quadrilateral

A) always

B) only if $$\frac{AD}{AB} > \frac{1}{2}$$

C) only if $$\frac{AD}{AB} = \frac{1}{2}$$

D) only if $$\frac{AD}{AB} < \frac{1}{2}$$

E) none of these

Not sure how to go about.

Any help will be appreciated. Thank you.

If $$\frac{AD}{AB}>\frac{1}{2}$$ we don't get a quadrilateral because $$GF$$ intersects with $$DE$$;
If $$\frac{AD}{AB}=\frac{1}{2}$$ so $$D\equiv G$$ and we don't get equilateral again.
If $$\frac{AD}{AB}<\frac{1}{2}$$ so it's valid.