This is not an answer but a comment. However, it will be too long and it would be awkward to break it up. It is of course natural to ask what is special about the nine point centre here. One can put the question in the following context. Given the shape of a quadrilateral $ABCD$, what is the shape of $PQRS$ (constructed as above but using any of the triangle centres protocolled in the online Encyclopedia of Triangle Centers)?
The shape of a quadrilateral is the unique pair $(p_1,q_1)$ and $(p_2,q_2)$ for which it is similar to the one with $(0,0)$, $(1,0)$ and these two points as vertices (in terms of complex numbers they are just $$ \frac{z_C-z_A}{z_B-z_A},\frac{z_D-z_A}{z_B-z_A}. $$
Many structural properties of a quadrilateral (in particular cyclicity) can be expressed as a simple equation in the $p$'s and $q$'s. For cyclicity, one equates the coordinates of the circumcentres of $ABC$ and $ABD$.
It is then a simple, if usually tedious, task to compute the shape of $PQRS$ (for a given centre function) in terms of that of $ABCD$ (easily automatised using Mathematica) and so provide far-reaching generalisations of Morley's result.
To be explicit, if $Z_1$ and $Z_2$ are $p_1+iq_1$, resp. $p_2+iq_2$, then we can easily compute the shape of $PQRS$ by computing the complex numbers which specify its vertices (using the centre function) and then forming the corresponding quotients as above.