$\angle PMQ= \angle PMN+ \angle QMN = \angle NPQ+ \angle NQP= \angle QNR= \angle QMR$ Done!

Hi, How did you go from angle PMN + angle QMN = angle NPQ + angle NQP Thanks

@above It follows from $P$ and $Q$ being tangency points of the two circles

I still don't understand, Would you please elaborate...

Just use the property: For a triangle $\Delta ABC$ with $TA$ being a tangent to it's circumcircle we have : $$ m(\angle TAC)=m(\angle ABC)$$

I understand now, thanks A lot!

$$\angle PMQ=180^{\circ}-\angle PNQ=\angle QNR=\angle QMR$$