Thales Application

euclidean-geometry

I'm struggling with this exercice:

Let $ABCD$ be a trapezium. $AB$ and $CD$ intersect at $O$.

$OE \parallel BD$, $OF \parallel AC $ and $F,C,B,E$ are aligned points

  • Prove that $EB=CF$

I think it's an application of Thales' theorem. I applied the theorem to the $\triangle$ s $OBF,OCE,OBC$ , but I don't see how to use it to prove that $EB=CF$.

I also noticed that we can prove that the triangles $OBE$ and $OCF$ are equal, $OE=OF$ , $OB=OC$, but I still need an equal angle in both triangles.
Thanks

Best Answer

You are right ! The problem does indeed involve Thales' theorem , and similarity:-

Note that we have:-

$\triangle OEC \sim \triangle DBC \implies \frac{EC}{BC}=\frac{OC}{DC} \tag{1}$ $\triangle OFB \sim \triangle ACB \implies \frac{BF}{BC}=\frac{OB}{AB} \tag{2}$

But by Thales' Theorem , the RHS of $(1)$ and $(2)$ are equal !

Therefore , $EC$ equals $BF$ , which implies $EB$ equals $CF$

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