Circle geometry with tangent line

Question

The circles $O_1$ and $O_2$ do not intersect and do not have the same radius.

$\overline{AB}$ and $\overline{CD}$ are tangent lines of circles $O_1$ and $O_2$, with $A$ and $C$ on the circumference of $O_1$, $B$ and $D$ on the circumference of $O_2$.

Let $\overline{EF}$ be the third tangent line of both circles, with $E$ on the circumference of $O_1$, and $F$ on the circumference of $O_2$,

Extend $\overline{EF}$ to intersect $\overline{AB}$ at $G$ and $\overline{CD}$ at $H$.

Prove that $\overline{GE}$ = $\overline{FH}$

I drew it like this:

I got $AB = CD$ but couldn't find any relations.

Sorry for my bad grammar.

Answer 1

$$GE=AE=AB-BE=CD-EF=CH+HD-(EG+GF)=$$ $$=HG+HF-EG-GF=(HG-GF)+HF-EG=2HF-EG$$ and we are done!

Answer 2

Tangents from a point to a circle are equal, so $GE=GA$, $GB=GF$, $HD=HF$ and $HC=HE\,$. Then:

$$\require{cancel} GE+GF = GA+GB = AB = CD = HD+HC = HF+HE \\ \implies \quad GE+GF=HF+HE \quad \iff \quad 2 GE + \cancel{EF} = 2 HF+ \cancel{FE} $$