>>12285616Euclidean geometry has already been solved by Tarski-Seidenberg, so unless you're interested in the computational efficiencies of geometric constructions, or an acolyte of Wildberger, you won't find much theoretical insight from it anyway.
>>12285622 >maybe it will give you a new perspective that people don't get doing analytic geometryI was going to say that this is the best reason to study geometry, aside from artistic purposes. But "gaining a fresh perspective" sounds pretty artistic as well.
And in the wake of the mechanization of geometry and automation of constructions, "artistic insights" may be the only way to advance the subject. I don't keep up with what they're doing at Erlangen, but Hilbert axiomatization (really, Godel's completeness theorem) didn't put an end to logic, nor did the Turing machine put an end to computability theory. If these are any indication, geometry will return to being an "applied math", with its development guided by advances in other subjects like cosmology, geography, and maybe even art.
>>12285541Turns out the proof isn't quite as simple as I expected: to prove , show that 0 is absorbing, then multiply both sides by -1, and then add 1.
