Geometry Proof Tutorial

Geometry problems used to not be my favorite math domain and I tended to prefer number theory or algebraic problems.  After starting a tricky problem my first thought was often "But if you just allowed the use of trigonometry you could bulldoze through this." But over this year I've looked at a lot of these problems and I've completely changed my mind about them. They make excellent puzzles and usually have multiple solutions, both important qualities. They also often make connections back into other fields like algebra. Based on practice I've also become a bit faster at working my way through them.

General Comments

  1. With more complicated problems it helps to break the problem into smaller pieces. Prove X that implies Y than implies Z.  You can work the problem in both directions. If I knew this then I'd be done so how do I figure that out?
  2. Fool around. Add in lines, look for relationships, experiment. Your first approach may not be correct so its important to be open to different lines of inquiry.
  3. Give the problem some time. For a hard problem I often go down the wrong avenue first and after a while set the problem aside for another day. When practicing you should always allow yourself enough time to think about the problem and not rush for hints or someone else's solution.
  4. Most problems have multiple solutions. So many of the approaches or examples I discuss can be done other ways (the geometric relationships manifest in multiple manners). I'm partial to techniques that apply across multiple problems.
  5. Observe the picture. If two parts (especially triangles) look congruent see if that can be proved to be the case. Don't be afraid to measure with a ruler or protractor to get some ideas.
  6. Its OK to backtrack as well. If you're proving an angle is x, its sometimes interesting to fill in all the angles based on that and see what it implies for example.

Basic Techniques

  1. Angle chasing: Look at all the angles and see which additional ones can be computed.

So for example in this figure above there are 3 isosceles triangles including the largest one. That means we can angle trace the top triangle OAC first. and then the lower own ACB as on right.

2.. Look at all the parallel lines next.
3. Add in targeted lines to connect points.

Find the shaded area if the radius of the circle is R.
   Generally speaking circle problems should always add in the center and connect it to all other points as so:

Now you have a wedge of the circle and a triangle both much more manageable shapes to  work with.

5. Search for congruent triangles using SAS, SSS, ASA et cetera.

More Advanced

6. Take advantage of similar triangles. Always do this first before #7.
7. Utilize the Pythagorean theorem.
8. Find two different expressions for the same angle/side/area.
9. Sometimes areas are easier to manipulate than  coordinates or sides. Example of two triangles
10. Area substitutions can be powerful.  You  can compose areas by both adding and subtracting

11. Consider rigid transformation i.e. rotation/translation/reflection of triangles or other shapes.
12. Create additional symmetry.

13. Look for an implicit circle.
14. Identify Cyclic Quadrilaterals. These are particularly powerful.
15. For equal angles consider the angle bisector theorem or extending to create a perpendicular bisector.
16. Don't forget about the alternate segment theorem.

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