I was looking at a solution with complex numbers to the following problem.
In $\triangle ABC$ with incenter $I$, the incircle is tangent to $CA,AB$ at $E,F$. The reflection of $E,F$ across $I$ are $G,H$. Let $Q=GH\cap BC$, and let $M$ be the midpoint of $BC$. Prove $IQ\perp IM$.
They claimed that $Q=DD\cap GH$ and proceeded to use the intersection of lines formula to get a complex number representation for the point $Q$. The rest of the solution worked perfectly and resulted in the perpendicular property stated in the problem.
How does this work? Because clearly the "degenerate line" $DD$ could be any line through $D$. Am I understanding something wrong?
Also, here is the diagram:
