First of all the game is quite nice, but there's a simple method with which you can win every level pretty fast, which kind of kills the fun as it is now.

How can I tell whether the circular sprit intersects with an arbitrary line.

I don't know how familiar you are with linear algebra, it really helps ;-)

(Update: When I talk about "points", I really mean "position vectors to points". Thanks FunkyMonk)

I'll call the end points of your line A and B, the center of your sprite M, the point on the line which is the closest to M is called C (all those are vectors), the radius of the sprite is r (a scalar value).

Since C is on the line between A and B, it must be expressible in the form A + x * (B-A). Since it's the closest point to M, the connection line from C to M is perpendicular to the original line, or as a formula:

(B-A) . (M-C) = 0
or
(B-A) . (M - A - x * (B-A)) = 0
# (that's a scalar product)