Proofs by contradiction are quite simple when writing proofs as prose. Make an assumption, derive a contradiction, and prove the assumption is therefore impossible. My question arises from the fact that we now have two different ways of constructing proofs, which include the 2-Column proof, which is seen in many geometry proofs.
How do we incorporate an assumption, or "for the sake of contradiction," in a 2-Column Proof?
I only see two options. I will use Euclid's Elements I.6, for example, as a classic Reductio ad Absurdum Proof.
1. Break the 2-column format momentarily to introduce the assumption.
| Statement | Reason |
|---|---|
| $\Delta ABC:\angle ABC=\angle ACB$ | Given |
For the sake of contradiction, assume $AB\ne AC$.
| more statements | more reasons |
| $\vdots$ | $\vdots$ |
$Q.E.D$
- Or, make the assumption in the statement column, and state "For the sake of Contradiction" as the reason.
| Statement | Reason |
|---|---|
| $\Delta ABC:\angle ABC=\angle ACB$ | Given |
| Assume $AB\ne AC$ | For the sake of contradiction |
| $\vdots$ | $\vdots$ |
$Q.E.D$
Neither option looks technically or mathematically correct insofar as introducing an assumption for the sake of contradiction in a 2-Column Proof.
Is there a proper formulation for introducing proofs by contradiction in the 2-Column Proof format? Are any of the proposed formulations here correct or incorrect?
