Can we ever prove a theorem cannot be proven directly (i.e. We must use contrapositive or prove by contradiction.)? Can we even rigorously defined whether a proof is direct or not?
Example:
I was reading Steiner–Lehmus_theorem: Every triangle with two angle bisectors of equal lengths is isosceles.
I knew a few elementary proofs of this theorem, but none of them are direct proof (for example: constructing congruent triangle).
The wiki page briefly mentioned the debate on whether equating algebraic expressions for angle bisectors and factoring is a direct proof or not, I don't quite understand that part either.
