Golden Ratio in Inscribed Equilateral Triangles

Tran Quang Hung has posted on the CutTheKnotMath facebook page a sighting of the Golden Ratio in inscribed equilateral triangles. He also supplied the proof.

Statement

Equilateral $\Delta DEF$ is inscribed into equilateral $\Delta ABC$ so that its extended midlines $MN, NP, MP$ pass through the vertices of $\Delta ABC, as shown:$

Then, say, the vertices of the inner triangle divide the sides of the outer triangle in the Golden Ratio, e.g., $\displaystyle\frac{BF}{AF}=\phi.$

Proof

Thales' theorem applied repeatedly gives the following sequence of proportions:

$\displaystyle\frac{AF}{BF}=\frac{MN}{BN}=\frac{FN}{AM}=\frac{BF}{AB},$

which is defining property of the Golden Section.