1 Nine triangles

We take nine congruent triangles (Fig. 1).

Fig. 1: The nine triangles

Now we rotate the red triangles until the upper vertices meet each other (Fig. 2).

Fig. 2: Where is the golden ratio?

The segments between the three blue points are in the golden ratio.

2 Three hexagons

We divide three regular hexagons into congruent halves (Fig. 3).

Fig. 3: Half hexagons

Now we rearrange the parts (Fig. 4).

Fig. 4: Golden ratio

The segments between the three blue points are in the golden ratio.

Acknowledgment

The author would like to thank Jo Niemeyer (Berlin) for helpful suggestions.