Prove that there always exists a line bisecting each of two point sets
We can prove this using the geometric dual of $P$ and $Q$.

Take the line arrangement of $P^*$ and $Q^*$ in the plane and let us look at the median levels $\Lambda_p$ and $\Lambda_q$. Their intersection is definitely a possible solution because $\Lambda_p$ is the point set describing the dual of all bisecting lines for $P$, and vice-versa for $Q$.

Note the following: Since $|P|$ and $|Q|$ are odd, both median line arrangements have the same slope (and since all slopes are unique by assumption, they must correspond to the same lines) in the beginning and the end of the arrangement. Let us denote $p^*$ and $q^*$ as the respective lines. Then at the beginning (i.e. the largest x-value where no intersections occured yet), one must be above the other and at the end (i.e. the smallest x-value such that no intersections happen from now on forward), they must have switched places.

Since the levels $\Lambda_p$ and $\Lambda_q$ are piecewise continuous (actually even continuously extensible), by the intermediate value theorem they must intersect in at least one point $c^*$, which is the dual of a possible simultaneously bisecting line $c$.