https://mathbitsnotebook.com/Geometry/BasicTerms/BTproofs.html

Originally, I was in search of how many different “types” of proofs there were, but I had a misunderstanding about proofs that this site helped me clear up.

There are different formats of proofs and different types of proofs. this site sheds light on the different formats.

Before reading this site, I had only used the Two-Column proof format. Euclid used a variety of types of proofs in “Elements, but almost always used the same format. There are four formats. You can find them here

Below is a description of each. In my Geometry class, I had only used the Two-Column proof, but I would have liked to use the others. I should note that these are all formats for a certain type of proof called Direct Proofs. We’ll get into the other types of proofs later.

The Two-Column Proof

This proof organizes the proof into two columns with “statements” on the left and “reasons” on the right. They are usually to be read and listed in chronological order of the argument. Thus, if you try to read the arguments out of order, it would lose its logical flow.

The Paragraph Proof

This format of proof has students organizing statements and reasons together in sentence form. It follows almost an exact format of a paragraph in an English class where there needs to be an introductory and a concluding sentence.

The Flow (Flowchart) Proof

This proof format uses a visual aid to show the statements and reasons why an argument is valid. This is unique because the order of the proof doesn’t matter as much as the two-column proof format and can be helpful for visual people.

and, the Transformational Proof

This proof format is one I was not aware of, but this is the type of proof in which Euclid used often. This format capitalizes on the ideas of rigid transformations — that is it makes sense of the argument by moving figures (either by translation, rotation or reflection) around by a specific set of rules.