These circles determine a point $F$, which corresponds to the (other) intersection of the two circles.
Let now prolongate the sides $AB$ and $BC$ in such a way that these prolongations intersect the two circles in $H$, $G$.
My conjecture is that the points $AFCGH$ always determine a circle.
Is there an elementary proof of such conjecture?
This post is related to this one A conjecture related to a circle intrinsically bound to any triangle.
I apologize in case this is an obvious result. Thanks for your help!