Supposing a triangle $ABC$, where $|AC|\neq |BC|$, denote the incentre $I$ and the points of tangency between the inscribed circle and $BC$,$CA$,$AB$ to be $D,E,F$ respectively. $M$ is the midpoint of $AB$. $K$ is the intersection of the perpendicular to $CM$ passing through $I$, and line $DE$. The task is to prove that $CK||AB$.
I have realized that $DE$ is a side of the Gergonne triangle of $ABC$, but I haven't been able to connect it with angle bisectors or anything pertaining to the incentre, nor with medians. I have also tried constructing the circumscribed circle and looking at the perpendicular bisectors but without much success.
I'd really appreciate your help.