Let points A and B be two points on the hyperbola, xy=1, and let C be the reflection of B through the origin.
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 $xy=1.$
This is the bit that seriously makes no sense.
"let C be the reflection of B through the origin."
Reflections happen across lines. I assume the line passes through the origin but whcih line are you talking about?
ALSO do not be so lazy when you post questions. Get rid of the meaningless $ and slashes. Make them are easily read as you are able.
Try using fact that there's a right triangle between ABA'. You're hunting for A' coordinates based off A or B coordinates.
Reflecting across origin is normal, disregard other dudes rants.
Please do not post solutions to this problem!
This is a homework problem, and the original poster is trying to cheat. I know, because I am in the same class, and have the same homework.