Cute Geo! Sketch: Firstly A-center everything. Let $BI \cap MN = Z$ and $CI \cap MN = Y$ where $M,N$ are midpoints of $AB,AC$ and $I$ is the incenter. We claim that $KP \cap LQ \in (YZ)$, which simply follows from the Iran lemma and a little angle chase.

@above, this is indeed the main idea, though the locus is just an arc of this circle since $X$ varies only on the segment $AC$.

a_507_bc wrote: @above, this is indeed the main idea, though the locus is just an arc of this circle since $X$ varies only on the segment $AC$. Yeah, I forgot to write that, we must specify the length of arc, just take $X \rightarrow B$ and $X \rightarrow C$ to get that if $DEF$ is contact triangle then midpoints of $DE,DF$ are the end points of arc.