Calculation:

We know that in similar triangles FGE and HGF,

\(\displaystyle\frac{{{F}{G}}}{{{H}{G}}}=\frac{{{G}{E}}}{{{G}{F}}}=\frac{{{F}{E}}}{{{H}{F}}}\)(by law of similarity)

Hence, we substitute the value of EH and HG in the equation above, we get

\(\displaystyle{G}{F}^{{2}}={G}{E}{\left({H}{G}\right)}\)

Thus, we get

\(\displaystyle{G}{F}={3}\sqrt{{{6}}}\)

We know that in similar triangles FGE and HGF,

\(\displaystyle\frac{{{F}{G}}}{{{H}{G}}}=\frac{{{G}{E}}}{{{G}{F}}}=\frac{{{F}{E}}}{{{H}{F}}}\)(by law of similarity)

Hence, we substitute the value of EH and HG in the equation above, we get

\(\displaystyle{G}{F}^{{2}}={G}{E}{\left({H}{G}\right)}\)

Thus, we get

\(\displaystyle{G}{F}={3}\sqrt{{{6}}}\)