Why are geometric proofs important?
You can prove something about geometry without the "I lost you at 'the square is a rectangle.'" kind of thing; you prove a law with multiple understandable steps.
Needing geometric proofs is kind of like needing lawyers to sue. You're not allowed to say, "You were a jerk and need to pay me," but rather "The defendant has committed fraud and laundering and has broken the law. He/she is legally obligated to pay the plaintiff." This makes sure that everyone is on the same page/speaking the same language.
Also, it's like speaking in English to an English-speaking person, even though you're more comfortable with Spanish. They might be more comfortable with German, but you can speak to each other.