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In the figure, ABCD is a parallelogram, $F$ is in the prolongation of $AD$. If $EF=32$, $GF=24$ and $AD=DF$, $BE=?$

I made the drawingenter image description here

My try: I tried to do some proportions between the triangles $FDG$ and $BCG$ but got nothing. I'm 100% sure that this problem is about propotionality. Any hints?

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  • $\begingroup$ What Kind of triangle is $$\Delta ABC$$? $\endgroup$ – Dr. Sonnhard Graubner Apr 7 '18 at 18:20
  • $\begingroup$ the problem doesn't say anything about that triangle. We don't know $\endgroup$ – Rodrigo Pizarro Apr 7 '18 at 18:26
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By intercept theorem, $BG=GF=24$

$BE=BG+GF-EF=24+24-32=16$

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  • $\begingroup$ wouldn't be $BE=16$? $\endgroup$ – Rodrigo Pizarro Apr 7 '18 at 18:36
  • $\begingroup$ @RodrigoPizarro The two numbers reversed. Thanks. $\endgroup$ – CY Aries Apr 7 '18 at 18:39
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Hint: Triangles DGF and ABF are similar.

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