Another way to get the golden ratio, using geometry and algebra

Today I was watching a video about how to construct a dodecahedron. In the first part of the clip, it was shown how to get the golden ratio with a specific type of geometric construction. I was absolutely intrigued by it and decided to figure out if the construction did indeed yield the result phi.

Remarkably and beautifully, I was able to confirm the result. My workings are shown below. I live for these eureka moments!

Part 1

Part 1

Part 2

Part 2

Part 3

Part 3 
Part 4
Part 4

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