Golden ratio and Fibonacci sequence

Diagonal of pentagon

pentagon diag

Each diagonal is parallel to the opposite side, so ABCD is a parellelogram. So CD=AB=1.
CE=\tau-1, and comparing sides of triangles BCD and ECF, we get
\displaystyle \frac{1}{\tau}=\frac{\tau-1}{1} \to \tau^2-\tau-1=0 \to \tau=\frac{1+\sqrt{5}}{2}.

See also phinary.

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