, 2023

Let points $A$, $B$, $C$, $A'$, $B'$, $C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$, $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the pedal triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular of $s(A')$ through point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear.