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I have not done a lot of problems in geometry. But, when I looked into the Olympiad questions and answers I could find that the solution to each question include drawing extra lines(drawing normals or extending a side of the given triangle etc.). How to successfully draw those extension to the geometric figures and to find the answer? How do problem solvers figure out whether there is any extra drawing required in the given geometric figure? Does it come through practice?

Rajath Radhakrishnan
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  • Its difficult to give specific advice as you have not given a specific example. But there are lots of geometry questions on this site such as here. Take a look at lots of them (particularly ones with accepted answers). Try to solve them your self then take a look at the accepted answer. It will come with practice. – Warren Hill Aug 31 '13 at 09:40

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If you want to solve an Olympiad problem, especially geometric one, you'll have to pracitce a lot. Practicing and solving a lot of geometric problems will make you familiar with all these theoremas and their application. Also that can better your creativity, because when you check one problem's solution (even if you don't solve it by yourself), you get an additional idea for a solving in future.

Every contest problem, especially Olympiad ones can't be solved using something you'll remember, which means that there isn't a general way to solve this one, it's all about your imagination and creativity to make use of the some easy and simple methods you already know. Actually if you have 40 participants, you'll almost surely end up with at least 10-15 distinct solution.

Problems in contest usually use some of the not so famous theorems, but yet they are relatively known to mathematicians. For exapmple: Ceva's theorem, Menelaus theorem, Power of point... If you need to solve a problem using these theorem, it won't be just simply application and also it won't be easy to identify the theorem you should use, because thay are somehow hidden inside the text. Note that the few theorems I mentiond above all need a line or point and it won't be given you have to draw it.

But once you solved a lot of examples, you are familiar with those theorems and you know what they can lead to and you'll know whether using that theorem will help you.

And finally here's an example. At one contest, there was a problem that needed use of the Simpson line. Simpson line was "semi-hidden", but if you have met it before, you'll almost surely have applied it, otherwise you'll lose precious time to prove something that is already known.

Stefan4024
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