The title seems ambitious, I know.
So, as evident, I would like to know that among the various types of approaches in Geometry, which one is the most "effective" in problem solving. Parameters are as follows:
1) Learning Curve: How difficult and vast is the whole thing? Do I need to have a terabyte of storage in my brain? Is there something 'special' required to learn it? How fast can I learn it's basics and start solving reasonably difficult problems?
2) Expanse: What types of problems can I solve? Are there difficulties in applying the method to some problems? Is it good for just computation? Or just proofs?
3) Power: How easily does it 'kill' problems? Can it make USAMO, IMO questions kneel? Can it convert a page of proofs/calculations into 2-liners?
4) Future Use: I'm in high-school now, so will learning a specific method come in use to me 5-10 years later? How useful will they be in scientific fields that are 'not directly' mathematics, like physics?
5) Coolness: (Optional) Do I feel like Iron Man when everyone uses some method for this unbelievably difficult problem and I kill it using my favorite method? Are the equations cool-looking?
I believe that those parameters quite present what I'm actually asking. Please present your opinions, 'with examples in support for your statements'. And for the love of God, don't close off the question as 'broad' or 'off-topic', because I've tried my best to narrow the question and get it 'on-topic'.
I personally love synthetic geometry, complex numbers but do not like cartesians, because, they don't seem very cool, so...
And, if there's some other form of solving geometry problems too, please do present them. I've left out 'higher' forms of geometry(differential, integral, combinatorial), basically, because... I don't know anything about them. And, trigonometry is taken for granted in all methods given.
Thanks a million for all your suggestions!!! Anything is appreciated! Literally, ANYTHING!