| How to Complete a High School Euclidean Proof Oct 12th 2012, 00:00 Euclidean geometry is one of the first mathematical fields where results require proofs rather than calculations. Proof-writing is the standard way mathematicians communicate what results are true and why. The entire field is built from Euclid's five postulates. Preparation Everything in this section is considered scratch work. It isn't part of the proof, but doing these steps will help you to write a correct and efficient proof later. It's hard to write a mathematical proof on the fly; you need to understand why it works yourself before you can communicate it in proof form. - Read the statement of the problem. Understand the definitions of all terms in both the givens and the proposed conclusion.
- Draw a diagram of the situation. Make all angles and distances as accurate and to scale as possible. Label all relevant points, angles, and distances. Note how each of the given assumptions manifest on the diagram.
- Redraw the diagram. Your first version will probably be unsuitable in some way. Maybe it was too cluttered to read clearly, maybe the intersection of so important a pair of lines is off the page, maybe you were told to assume that three angle bisectors of a quadrilateral intersected in a single point, and that doesn't happen with the one you drew. In any case, you learned something from the first attempt that will make your next attempt better.
- Make observations from the diagram. Do two lengths look equal? If so, can you prove it? What plausible hypotheses, if true, would help you derive the intended conclusion? Write down any relationships between various parts of the diagram that you can derive from your assumptions. Note, this is where an accurate diagram helps. If two angles look unequal, then you know that no correct proof will involve the assertion that they are equal. With an inaccurate diagram, you never know.
- Recall or look up any previous results that might help. It's very common for mathematical results to depend on previous work. Hint: If a theorem has a name (Pythagorean Theorem) or an abbreviation (CPCTC), it's probably used in very many later results and you should make sure you understand it.
- Work backwards too. Try to guess the second to last line of the proof. If you're trying to show that the areas of two triangles are equal, what would you need? Maybe they're congruent, but that's a much stronger result. If an edge of one is congruent to an edge of the other, then can you show that the corresponding altitudes also have the same length?
- When you've discovered a way to logically link the initial conditions with the conclusion, make a proof sketch. Highlight the important intermediate steps and the major theorems needed to derive them.
Formal Proof Once your background work is sufficient, it's time to convert that into a formal proof. - Draw a diagram. This doesn't have to be overly accurate––formally, it isn't required at all, but it usually helps. Name all points, angles, or other features that you intend to refer to in the proof later.
- State the theorem. State the given assumptions, and what you intend to conclude from them.
- Set up the format for a two column proof. Label the left column Statement, and the right column Reason.
- Repeat each of the givens for the first few rows of the proof. For the reason, write "given." Even if some of the givens aren't needed until later, it can't hurt to list them early.
- Aim towards the first important intermediate result you found in the preparation stage. Write each step towards that result, and justify each with an appropriate reason. The types of acceptable reasons are small. They include:
- Given
- Definition
- Axiom (or postulate)
- Previously proven theorem (or lemma, formula, law, etc.).
- If the reason is a theorem, be sure to specify which one and why it applies. The following methods are usually acceptable:
- Restating it. (The three altitudes of a triangle intersect in a single point.)
- Referring to it by name. (Thales' theorem.)
- Reference to a text. (Theorem 5.3 on page 124.)
- By a standard abbreviation. (SAS).
- If a theorem has conditions attached to it, you should explicitly show how they are satisfied. For example, if your statement is that triangle ABC is congruent to triangle DEF, you might use this detailed reason: BC = EF (line 5), <ABC = <DEF (line 13), <BCA = <EFD (line 16), and the ASA congruence theorem. Here the earlier 5th, 13th, and 15th lines of the proof show the prerequisite angle and side congruences for ASA to apply.
- Continue working towards your conclusion by establishing the other key intermediate steps you found in the prep work. Make sure that each step follows from the previous ones.
- The last line of the proof should be the desired conclusion. As with all other steps, justify it with an appropriate reason.
- Optionally, terminate the proof with QED, a box, or some other similar mark.
Edit Warnings - Be alert for any hidden assumptions. There is a famous fake proof sometimes attributed to Lewis Carroll that all triangles are isosceles.[1] The error is subtle when presented like this. All triangles alleged to be congruent are, indeed, congruent, and for the reasons stated. There is a hidden assumption implied by a misleading diagram. An accurate diagram will reveal the false assumption, and show why the proof goes bad.
- In introductory geometry courses, it's common to require the use of two-column or similar proof formats where each step is formally justified. In more advanced work, this is less common as excessive rigor can distract from the important ideas. Still, it is expected that an informal paragraph proof could, upon demand, be converted into a fully rigorous proof with an explicit justification for each step.
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