A quadrilateral ABCD is inscribed in circle Q, and its diagonals intersect at point P. ABCD has no parallel sides. Let M be the intersection of $\overline{AB}$ and $\overline{CD}$ and let N be the intersection of $\overline{AD}$ and $\overline{BC}$. Let $\overline{MP}$ intersect Q at points F and G. Prove that the tangents to Q at points F and G intersect at N.
Well if you know polar theory then this is pretty simple.
Line $MP$ is polar line for point $N$.
Since $F$ lies on polar for $N$ then $N$ must lie on polar for $F$ which is tangent at $F$. Since $G$ lies on polar for $N$ then $N$ must lie on polar for $G$ which is tangent at $G$.
Thus conclusion.
You can look here: http://www.imomath.com/index.php?options=334&lmm=0 and: https://en.wikipedia.org/wiki/Pole_and_polar