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.
 A: Depending on the education you've had, the difference is likely basically that between your school algebra and geometry courses. In a first geometry course you usually get a list of axioms for points and lines (a straight line segment can be drawn between any two points ,etc.) Then you create proofs of properties of figures in the plane using these axioms, without ever writing down any algebraic equations. Specifically, you never use coordinates on the plane in doing this sort of geometry, which is what is called synthetic. 
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.
