1 Solution with 7 circles

Step by step:

Step 1

Step 2

Step 3

Step 4

The golden section arises

The distances between the black point and the two red points are in the ratio of the golden section.

Proof 1:

Proof Figure

Let the first circle be the unit circle with radius 1. Then we have:

The number is the golden section (see [Walser 2001], p. 4)

Proof 2:

Embedding in the Figure of Odom (see [Walser 2001], p. 123)

Figure of Odom

2 Solution with 5 circles only

See [Hofstetter 2002]. Step by step:

Step 1

Step 2

Step 3

The distances between the black point and the two red points are in the ratio of the golden section. The proof is left to the reader.

Reference

[Hofstetter 2002] Hofstetter, Kurt: A Simple Construction of the Golden Section. Forum Geometricorum, 2 (2002), p. 65-66.

[Walser 2001] Walser, Hans: The Golden Section. Translated by Peter Hilton and Jean Pedersen. The Mathematical Association of America 2001. ISBN 0-88385-534-8