$A B C D$ is a convex quadrilateral. How many points are there in the plane that, when reflected successively over the lines $A B, B C, C D$ and $D A$, finally return to their original position?
It is known that the successive reflection over two lines is either a rotation or a translation. If the lines $e$ and $f$ intersect at point $M$ and the angle between them is $\alpha$, then the $2\alpha$-degree rotation about $M$ is equivalent to the successive reflections over $e$ and $f$ (Figure 1); if the two lines are parallel and the distance between them is $d$, then the result of the successive reflections over the lines is a translation perpendicular to the lines, with a distance of $2d$ (Figure 2). The direction of the rotation or translation depends on which of the lines $e$ and $f$ is reflected over first.

Therefore, the reflections over $AB$ and then $BC$ can be replaced by a $2\angle ABC$-degree rotation about $B$, and the reflections over $CD$ and then $DA$ can be replaced by a $2\angle CDA$-degree rotation about $D$. These rotations can be replaced by two successive reflections over axes, and we can assume that one of the axes is the line $BD$ in both cases. Let $g$ be the line through $B$ and $h$ be the line through $D$, such that the successive reflections over $g$ and $BD$, and over $BD$ and $h$ are equivalent to the $2\angle ABC$-degree rotation about $B$ and the $2\angle CDA$-degree rotation about $D$, respectively. The lines $g$ and $h$ can be constructed simply by considering the angle equalities shown in Figure 3. Thus, the successive reflections over the four sides of the quadrilateral $ABCD$ can be replaced by the successive reflections over $g$, $BD$, $BD$ again, and $h$. However, this is the same as reflecting over $g$ and then $h$, because reflecting over the same line twice in succession is the identity transformation. Therefore, our problem is reduced to finding how many fixed points the successive reflections over $g$ and $h$ have.

The successive reflections over two lines are a translation if $g \parallel h$. This happens precisely when $\angle ABC + \angle CDA = 180^\circ$ (Figure 4), which means that $ABCD$ is a cyclic quadrilateral. A (non-identity) translation has no fixed points, so in this case, there is no point in the plane that returns to its original position after the transformations. If $g$ and $h$ intersect at point $P$ (Figure 3), then the transformation is a rotation about $P$ by an angle less than $360^\circ$, which has $P$ as its only fixed point.

In summary, there is either no point or exactly one point in the plane that returns to its original position after the successive reflections over the lines $AB$, $BC$, $CD$, and $DA$, depending on whether $ABCD$ is a cyclic quadrilateral or not.
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