φ and Fibonacci

Mathematicians have long been fascinated by a particular number, called the Golden Ratio. For an excellent read about the properties and surprises of the Golden Ratio, pick up a copy of this book by Mario Livio. There are a lot of equivalent ways to define the golden ratio; let's use the following definition:
For any segment AB, there is a unique point C on AB such that
Let's find the value of !
Let x be the length of , and let y be the length of .
By the quadratic formula, we get:
Note that both AC and BC have positive lengths (of course), so
But , so we must have
We have found two properties of :
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