Golden Window

Golden Ratio made multiple appearances at the house of Jerzy Kocik (Golden Window, Math Magazine, 83 (2010) 384-390)) who enjoys a window of a peculiar design. Start with two small central circles of unit diameter. The radius R of the two circles on their left and right, given that a pair of congruent circles (dotted) is simultaneously tangent to all the other circles, is exactly φ. As the diagram below shows various powers of φ, too, pop now and then:

golden ratio from two equal circles - a wonderful window

In a two-dimensional configuration with multiple sightings of the golden ratio one would expect to find some occurences of the golden rectangle. Indeed, there are a few in a circular design:

golden rectangle in a golden window

There are several golden rectangles in the semicircular design as well:

golden golden rectangle in a semicircle

To cap it off, there is an instance of Kepler's triangle:

Kepler's triangle in a semicircle

In addition, two more inscribed circles are perfectly aligned in the configuration:

two more circles in the goldon window

The latter feature could be easily verified. First, with the help of the Pythagorean theorem, we find the radius of the large inscribed circle (the one above the two initial ones) to be R = (2φ + 1)/(2φ - 1). Next check the alignments. If r is the radius of the two new circles, then the horizontal alignment requires

1 + φ - r = (2φ + 1)/(2φ - 1).

The vertical alignment imposes another condition:

φ + r = φ³ - (2φ + 1)/(2φ - 1).

The third condition comes from the new circles being tangent to the big one:

(1 + φ)² + (r + φ)² = (φ³ - r)².

But, since

φ³ = φ×φ² = φ(φ + 1) = φ² + φ = (φ + 1) + φ = 2φ + 1,

all three conditions are seen to be actually the same. From either, we get r = φ/(2φ - 1).

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