Golden Ratio In Rectangles

Two recangles reveal directly the presence of the Golden Ratio: $1\times 2\;$ and $1\times 3\;$ rectangles. Tran Quang Hung has posted both cases at the CutTheKnotMath facebook page.

$1\times 2\;$ rectangle

golden ratio in a circle and a 1x2 rectangle

and

$1\times 3\;$ rectangle

Gloden ratio with a circle and a 1x3 rectangle

In both cases the ratio of the blue segment to a red one is Golden. Quang Duong has observed that the two diagrams neatly overlay, showing that the same segments may be seen as belonging to either case:

Gloden ratio with a circle and both 1x3 and 1x2 rectangles

Note

The construction in a $1\times 3\;$ rectangle happend to be a part of a more complicated diagram with multiple occurences of $\varphi\;$ that has been posted by Elliot McGucken earlier in multiple places, including CutTheKnotMath facebook page.

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