A construction mess (Euclidea Series)

Part of my motivation for writing this series is to make a confession: some of my constructions are just a mess built of geometrically calculating a length that I've determined algebraically. Typically, my approach is to create a coordinate system, then set up a couple of equations that determine a key length for the construction, solve those equations, then use geometric operations to construct that value.

Here is an example. Theoretically, it is a spoiler for a construction in mu pack (circle tangent to two circles 12.5) but I doubt anyone will really be able to see what is going on here:

Another example is construction of the regular pentagon. I know the golden ratio figures prominently, so one approach I often use is just to build that ratio between two lengths and then impose it on the basic construction template.

Other than massively overrunning the move targets for E and L stars, the weakness of this approach is that it doesn't link with any geometric insight about the construction. In a sense, this is the power of analytic geometry: you can get results without having to find a new insight.

Over the course of the series, I'll try to work on constructions that are geometrically insightful, but won't shy away from letting you know where I've had to push through with a brute force attack.