Line-Segments In Circle

Here is a game of mine that is NOT based on an n-by-n grid...
The game is for 2 players.
Draw a circle on a piece of paper. (Preferably the circle is drawn
with a compass, but this is not necessary. The circle should be drawn
carefully, however.)
Put a dot at the circle's center.
Players take turns drawing straight line-segments (preferably with a
straight-edge) within the circle as follows:
*A segment can go from the dot at the circle's center to the edge of
the circle.
*A segment can go from {an intersection of another segment and the
circle} to {another intersection of a segment and the circle}.
*A segment can pass through at most ONE line-segment that has already
been drawn.
*A segment can not coincide with a previously drawn segment. (ie. Two
segments can coincide at at most one point.)
*A segment cannot be drawn from the center to the circle that is 180
degrees away from the segment last drawn by the other player, if the
last segment was drawn from the center to the circle. (This prevents a
player from simply matching the other player's moves, and so
automatically winning.)
The last player able to move is the winner.
Is there a strategy that makes this game easy to win?
Thanks,
Leroy Quet