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$D, G$ 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.

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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.

Now, we see that A) and E) are not relevant.

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