What Is Proof by Contrapositive?

To understand proof by contrapositive we must first understand what the contrapositive of a statement is.

Say there is a statement that says: If a then b.

Then the contrapositive of this statement is : If not b then not a.

Contrapositive proofs are designed on the fact that if the contrapositive of a statement is true then so is the statement.

Here is an example:

Prove that if a, b, and c are not sides of a right triangle and c is the longest side, then $a^2+b^2\ne c^2$.

Let's take the contrapositive, this yields: if $a^2+b^2 = c^2$, then a, b, and c are sides of a right triangle where c is the longest side.

This is just the converse of the pythagorean theorem (Which is true), and so therefore the original statement was true.