How to Complete a High School Euclidean Proof

Step-by-Step Guide

1. 1

   Step 1: Read the statement of the problem.

   Understand the definitions of all terms in both the givens and the proposed conclusion.
2. 2

   Step 2: 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. , Your first version will probably be unsuitable in some way.
   
   Maybe it was too cluttered to read clearly, maybe the intersection of an important 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. , 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. , It's very common for mathematical results to depend on previous work.
   
   Hint:
   If a theorem has a name (like the Pythagorean Theorem) or an abbreviation (CPCTC for "Corresponding Parts of Congruent Triangles are Congruent"), it's probably used frequently in later results so make sure you understand it. , 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? , Highlight the important intermediate steps and the major theorems needed to derive them.
3. 3

   Step 3: Redraw the diagram.

4. 4

   Step 4: Make observations from the diagram.

5. 5

   Step 5: Recall or look up any previous results that might help.

6. 6

   Step 6: Work backwards too.

7. 7

   Step 7: When you've discovered a way to logically link the initial conditions with the conclusion

8. 8

   Step 8: make a proof sketch.

Detailed Guide

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