Let \(A\) and \(B\) be two points on the hyperbola \(xy=1,\) and let \(C\) be the reflection of \(B\) through the origin.
(a) Show that \(C\) is on the hyperbola.
(b) Let \(\Gamma\) be the circumcircle of \(\triangle ABC\) and let \(A'\) be the point on \(\Gamma\) diametrically opposite \(A.\) Show that \(A'\) is also on the hyperbola \(xy=1.\)
(no images were given with this problem)
Thank you very much for your help!