If you look at the "rules of geometry" and look for real world applications. Where do people use those methods?

Use a wire to get a straight line? Used in construction

Use a determined length of wire to compare lengths ? Used in construction

Use a determined length of wire to draw circles? Used in construction

Use 2 lengths of wire to construct parallel lines? Used in construction

Use lengths of wire to construct a perfect 90 degree angle? Used in tool-making (ie. to make a jig that you then use to measure)

The list goes on and on and on. If you've never used CAD packages, you should. The amount of geometry that goes into part design in everything from construction to "tools and construction engineering" as it's called in my country is incredible. Incredible as in they reserve 8 months in university to make sure people who already would have had pretty difficult math in high school are up to date on advanced geometry (plus teaching them how to properly extend what they know into 3d)

There's been updates since Greek times, such as using lasers for drawing straight lines instead of wires, but not as many as you'd think.

This is great because as anyone will point out, doing things this way (with correct equipment) results in accuracy across large distances (up to dozens of kilometers) that you just cannot match with analytical geometry + measurements.

Geometry was obviously invented and fine-tuned for thousands of years, for construction. That's what it was used for 4000 years ago ... and that's close to the only reason to still use (non-analytical) geometry today.