Connection between algebraic geometry and high school geometry. if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians are working on compass and ruler constructions anymore.
However: A lot of people seem to be working or at least use techinques from an area of mathematics known as algebraic geometry (in fact: today the abel prize was given to an algebraic geometer).
My question is the following:are the same kind of mental abilities required to understand and solve problems in algebraic geometry or are they more similar to other areas such algebra, topology and combinatorics?

While the following book: Elementary Algebraic Geometry by Klaus Hulek (American Mathematical Society, Volume 20, Student Mathematical Library), 2003 is not aimed at high school students (rather I would say about the level of mathematics major juniors in college) looking at it might give you some idea of the kind of concerns and issues, connections if you will, that this subject tries to make between algebra and geometry.
