a paragraph proof
When you explain to a friend why something must be true, you do not draw a two-column table — you talk it through in sentences, one reason flowing into the next. A paragraph proof is exactly that: a complete, rigorous argument written as ordinary connected prose.
A paragraph proof contains all the same ingredients as a two-column proof — the given information, the chain of justified statements, and the conclusion — but they are woven into flowing sentences instead of a grid. Each claim is still backed by a reason (a definition, postulate, theorem, or property), signalled by words like 'because', 'since', 'by the segment addition postulate', 'therefore', and 'so'. The art is to make the logical order plain in prose: start from the hypothesis, justify each move as you go, and arrive at the conclusion without gaps.
Paragraph proofs are how nearly all real mathematics is written, from textbooks to research papers; the two-column form is mostly a learning device. Writing in prose forces you to make the connective tissue explicit rather than hiding behind a column heading. The danger to guard against is the opposite of two-column rigidity: a paragraph can drift into hand-waving, where words like 'clearly' or 'obviously' stand in for a reason that was never actually given. A good paragraph proof reads smoothly yet leaves no step unjustified.
Given that B is the midpoint of AC, we prove AB = (1/2)AC. Because B is the midpoint, AB = BC by the definition of midpoint. Since B lies between A and C, the segment addition postulate gives AB + BC = AC. Substituting AB for BC, we get AB + AB = AC, that is 2·AB = AC. Dividing both sides by 2 yields AB = (1/2)AC, as required.
The same logic as a two-column proof, written as connected sentences with reasons in words.
Flowing prose is no excuse for hand-waving: 'clearly' and 'obviously' are not reasons. Every step still needs a stated justification, just expressed in sentences.