Eh, might as well just finish this one:
Let’s first set \angle QAR=2x, \angle QOR=2y, \angle COQ=2q and \angle ABC =2z (the goal is to now arrive to the conclusion that x=z.
From here we can derive a plethora of angles:
\angle PBR=2x (same arc with respect to big circle)
\angle CBR=q+y (inscribed angle of small circle)
\angle CBP = q+y-2x (simple subtraction)
\angle ARP=y+q (inscribed tangent theorem - perhaps I have the wrong name)
[it states that \angle ARP is equal to half the angle of arc CR]
Now we know that APRB is cyclic.
Thus \angle ABP=\angle ARP,
This means that 2z+q+y-2x=y+q.
This simplifies to z=x and we are done!
‘I have seen the leaderboard and will be finishing this question now’
How do you make those diagrams ?