Let $\Gamma$ be a non-degenerate conic section. Find all points $P$ such that there exists $3$ distinct points $A,B,C$ on $\Gamma$ such that $P$ is the incenter of $\Delta ABC$.
I came up with it myself but I have no idea how to solve this problem. I can only set $A(x_A,y_A),B(x_B,y_B),\ldots$ and then use the formula for the cartesian coordinates of incenter, but this does not seem to lead to any solution.