a two-column proof
A good argument is like a staircase: each step rests on the one below, and anyone can see why it holds. A two-column proof is a tidy way of writing such an argument so that every claim sits beside the reason it is allowed.
The page is split into two columns. The left column lists statements — the claims of the argument, one per line, starting from what you are given and ending at what you set out to prove. The right column lists, for each statement, the reason that justifies it: a given fact, a definition, a postulate, a previously proved theorem, or a property of equality. You may write a statement only if you can name a legitimate reason for it. The proof of a conditional 'if hypothesis, then conclusion' starts from the hypothesis (the 'given') and proceeds by justified steps to the conclusion.
The two-column format is a teaching scaffold rather than the only style mathematicians use; working proofs are usually written as flowing paragraphs. But the discipline it instils is real and permanent: every assertion needs a justification, and you cannot skip from premise to conclusion by intuition alone. A frequent beginner's error is to write a true statement with a vague or missing reason — in a two-column proof, an unjustified line is simply not a proof, no matter how obviously true the statement looks.
Given: B is the midpoint of AC. Prove: AB = (1/2)AC. Statement 1: B is the midpoint of AC (Reason: Given). Statement 2: AB = BC (Reason: definition of midpoint). Statement 3: AB + BC = AC (Reason: segment addition postulate). Statement 4: AB + AB = AC, i.e. 2·AB = AC (Reason: substitution). Statement 5: AB = (1/2)AC (Reason: division property of equality).
Every statement on the left is paired with a named justification on the right.
A statement without a valid reason is not a step in a proof, however obvious it seems; the two-column format makes that requirement impossible to dodge.