The ringmaster of a flea circus puts four fleas $A$, $B$, $C$, $D$ on four different points in the plane that form the corners of a square.
- Whenever the ringmaster shouts "Hop!", one of the four fleas jumps over one of the other fleas to the mirror point on the other side. (In other words, a flea sitting in point $x$ may jump over a flea sitting in point $y$ to the new point $z$, so that $y$ is the midpoint between $x$ and $z$.)
- While the fleas are jumping around, sometimes two of them may be sitting simultaneously on the same point. (This is fine, as these fleas are infinitesimally small.)
Question: Is it possible that after some time the three fleas $A$, $B$ and $C$ are sitting in points on the same straight line?