Can you apply proof by contrapositive on proof by contradiction In a proof by contradiction, a false statement implies a contradiction. What happens if you take the contrapositive of this? A not-contradiction implies a not-false statement. I.e. a tautology implies a true statement. Is this right?

Both are forms of indirect proof. As you would expect, a proof by contradiction makes use of a contradiction. A proof by contrapositive does not.
You can use proof by contradiction to infer that a statement $X$ is true by first assuming $X$ is false and then deriving a contradiction of the form $Y \land \neg Y$ or $Y\iff \neg Y$. 
You can use proof by contrapositive to infer that an implication of the form $X\implies Y$ is true by first assuming $Y$ is false and then proving that $X$ is false. No contradiction is required.
Of course, you can use both methods several times in the same proof. 
