Given a triangle $\Delta ABC$ and a point $P$, we define $P_A, P_B, P_C$ as the reflections of $P$ around $BC, AC, AB$ respectively.
Now, $P_A, P_B, P_C$ are collinear if and only if $P\in(ABC)$. (1)
Furthermore, define $c$ to be a circle concentric with $(ABC)$, i.e. $c$ has the circumcenter of $\Delta ABC$ as its center.
Then $A_{\Delta ABC}$ stays invariable as $P$ varies along $c$. (2)
I managed to prove (1) (using a homothety of the Simson line), but how can you prove (2)?
I'm especially interested in a synthetic proof.