What kind of geometry is useful to study for mathematical competitions?

Question

I'm bad in geometry but I would like to be better. What kind of geometry is useful to learn olympiad level geometry? I mean, can projective geometry solve more problems than geometry with complex numbers or analytical geometry? And which of those geometries allows me to write shortest proofs on average as there is a time limit in competitions?

Answer 1

The answer depends on what you mean by "bad in geometry." For many people the problem is very simple. Geometry isn't taught in schools the way it once was, so you can be very good at algebra and other things, but geometry can still seem foreign and strange.

It might be good to learn geometry - at first - in normal school books from earlier times in order to have a good base, before moving on to more difficult contest-level problems. What the best book is may depend partly on what languages you can read. For English, have a look here: http://www.knowledge-dojo.com/ , particularly Durell's A New School Geometry.

Answer 2

I'd recommend learning some non-Euclidean geometry as well. Pretty much master geometry with complex numbers and analytic geometry. The main idea at these mathematical competitions is to totally master your emotions and any nervousness you may have. Remember that you've trained well!