Cute parmenides51 wrote: Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear. Note that $IB, ID$ are isogonal in angle $AIC$ and $IA, IC$ are isogonal in angle $BID$. Since the line through midpoints of $AC$ and $BD$ passes through $I$, its reflection in the common bisector of angle $AIC$ and $BID$ passes through $X$ and $Y$, as desired.