# What is synthetic geometry?

What is synthetic geometry? Could you provide a short (i.e. a paragraph or two, not much longer) explanation in general elementary terms? In particular, I hope to be able to understand the contrast between synthetic and analytical (invariant-theoretic?) approaches in the end.

Wikipedia's definition is a bit too advanced for me.

• Have you tried Wikipedia? Commented Feb 9, 2014 at 1:00
• I assume you want an explanation that differs from wikipedia. Or perhaps wikipedia is too advanced? Please be more specific. Commented Feb 9, 2014 at 1:00
• @rghthndsd Wikipedia's definition is too advanced. I don't really understand the difference between analytical and synthetic as it is discussed in that article. Commented Feb 9, 2014 at 1:01
• do you know how to do geometry with coordinates? Commented Feb 9, 2014 at 1:08
• "Synthetic" just means axiomatic in this context. Commented Feb 9, 2014 at 1:49

The ancient Greeks took synthetic geometry in the plane and 3-dimensional space to an amazing level, much further than almost anyone learns today. For example, they understood the "conic sections," that is, parabolas, ellipses, (circles,) and hyperbolas, almost completely. But today you usually think of a parabola, for instance, as being the graph of an equation $y=ax^2+bx+c$. It's this equation, which requires choosing a pair of coordinate axes on the plane and a unit length on each axis, that's what analytic geometry uses that synthetic geometry avoids. The synthetic-analytic distinction is still very visible in modern geometric research, even though the particular questions of interest have changed drastically.