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:

golden ratio by Tran Quang Hung, construction #7

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

Using Thales' theorem and similarity of triangles we obtain 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.

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