Let $K$ be an inversion circle with center $O$ and let $C$ be the point of intersection of two lines tangent to $K$ in $A$ and $B$. Then let $E$ be the intersection of the line $AB$ and the line tangent to $K$ in another point $D$. Prove that the line $CD$ is the polar to $E$.
So I know that the intersection between $AB$ and $OC$ is the inverse point of $C$, so I think I have to use that in order to find another point $F$ in $K$ such that $FE$ is tangent to $K$, therefore the intersection between $DF$ and $OE$ is the inverse point of $E$. However, from there I don't know how to relate both ideas in order to get that $D$, $F$ and $C$ are collinear. Any help?
You can use the fact that reciprocation (exchanging poles and polars) preserves incidence. If a point is incident with a line, then the polar line of that point is incident with the pole of the line. If three lines are concurrent (i.e. meet in a point) then their poles are collinear.
Writing $t_A$ for the tangent in $A$, you start by observing that $t_A, t_B, CD$ meet in $C$. So reciprocate them all and you know that $A$, $B$ and the pole of $CD$ are collinear. So the pole of $CD$ must lie on $AB$ (as we just showed) and on $t_D$ (since that is part of the classical construction of a pole), so it is in fact the point $E$.