Golden Ratio in Regular Pentagon

The golden ratio, $\phi=\displaystyle\frac{1+\sqrt{5}}{2},$ makes frequent and often unexpected appearance in geometry. Regular pentagon - the pentagram - is one of the places where the golden ratio appears in abundance.

To mention a few (some of which have been proved elsewhere, others are straightforward):

$\displaystyle\frac{DE}{EX}=\frac{EX}{XY}=\frac{UV}{XY}=\frac{EY}{EX}=\frac{BE}{AE}=\phi.$

Most recently Dao Thanh Oai posted an observation at the CutTheKntMath facebook page that in the following diagram $\displaystyle\frac{FB}{FA}=\phi ,$

for $EF\perp DE.$ An easy proof is obtained by angle chasing. Observe that $\angle AED=108^{\circ},$ making $\angle AEF=18^{\circ}.$ Now draw $EB.$

Note that $\angle BEF=18^{\circ}$ also, from which $EF$ is the bisector of $\angle AEB.$ Using the property of angles bisectors, $\displaystyle\frac{BF}{AF}=\frac{BE}{AE}=\phi.$

If $F'$ is t he intersection of $DE$ and $AB,$ then also $\displaystyle\frac{BF'}{AF'}=\phi,$ since $EF'$ is the external angle bisector in $\Delta ABE.$

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