Let $ V $ $ \equiv $ $ AC $ $ \cap $ $ BD $ and let the tangent of $ \gamma $ passing through $ N $ cuts the tangent of $ \odot (CVD) $ passing through $ V $ at $ X $. From $ \triangle DAP $ $ \sim $ $ \triangle DQA, $ $ \triangle CBQ $ $ \sim $ $ \triangle CPB $ we get $ \tfrac{{CV}^2}{{DV}^2} $ $ = $ $ \tfrac{{CB}2}{{DA}^2} $ $ = $ $ \tfrac{CP \cdot CQ}{DP \cdot DQ} $, so $ VP, $ $ VQ $ are isogonal conjugate WRT $ \angle CVD $ (Steiner theorem), hence $ \odot (PVQ) $ and $ \odot (CVD) $ are tangent to each other at $ V $. Since $ VX $ is parallel to the tangent of $ \gamma $ passing through $ M $, so $ XV $ $ = $ $ XN $ $ \Longrightarrow $ $ X $ lies on the radical axis $ CD $ of $ \odot (CVD) $ and $ \gamma $, hence from $ {XN}^2 $ $ = $ $ {XV}^2 $ $ = $ $ XP $ $ \cdot $ $ XQ $ we conclude that $ \odot (PNQ) $ is tangent to $ \gamma $ at $ N $.