First we notice that the given length conditions uniquely determine two Apollonian circles wrt $A,B$ and $B,C$. Thus, the only possible locations of $P$ and $Q$ are the intersection points of these circles, which are obviously non-coincidental and uniquely determined in a given triangle. These are the only two points which can have the given properties (by the definition of an Apollonius circle). Consider an inversion wrt the circumcircle of $\triangle ABC$ ($r=OA=OB=OC$, where $O$ is the circumcentre). Suppose it maps $P$ to $P^*$. As $A,B,C$ stay fixed under the inversion, we can calculate the distances $AP^*,BP^*,CP^*$ by the inversion distance formula. $$AP^*=\frac{r^2}{OP \cdot OA}\cdot AP=\frac{r}{OP}\cdot AP$$Analogously, $BP^*=\frac{r}{OP}\cdot BP$ and $CP^*=\frac{r}{OP}\cdot CP$. Thus, we can notice that $$\frac{AP^*}{BP^*}=\frac{AP}{BP}=\frac{21}{20}, \quad \frac{BP^*}{CP^*}=\frac{BP}{CP}=\frac{20}{19}$$so the point $P^*$ satisfies the same length conditions as $Q$ and is different from $P$. By the uniqueness in the first paragraph, we can conclude that $P^*=Q$. This implies that $P$ and $Q$ are inverses under the inversion; as the centre of inversion is $O$, the three points are collinear. $\square$