Disclaimer: The solutions we've shared are just one exciting approach, and there are surely many other wonderful methods out there. We’d love to hear your alternative solutions in the community thread below, so let's keep the creativity flowing!

Firstly we have $\angle EAH = \angle EGH = 90$, hence $AEGH$ is cyclic. We now have, $\angle BHA =\angle GEB = \angle CEB $. Hence $\triangle BAH \cong \triangle BEC$ (ASA Congruency). So $AH = BF$ so $AHFB$ is a rectangle.

Now, we have $\angle HGC = \angle HFC = \angle HDC = 90$.  So, $HGFCD$ pentagon is cyclic. Also, $\angle BGP = \angle BFP = 90$. So $BGPF$ is also cyclic. 

Now, notice the fact that the radical axises of circles $(HGFCD),(BCD),(BGPF)$ have to be concurrent. Two of these radical axises are $CD$ and $GF$ which are concurrent at $Q$. So the third radical axis or the radical axis of $(BCD),(BGPF)$ must pass through $Q$. Let $R = DP \cap (BGF)$. 

Now we have 

\[\angle BRD = \angle BRP = \angle BFP = 90\ = \angle BCD \]

This implies that, quad BRCQ is also a cyclic quad . So, $BR$ must be the radical axis. So $B,R,Q$ are collinear. Hence we are done