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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 Γ be the circumcircle of triangle ABC and let A' be the point on Γ diametrically opposite A. Show that A' is also on the hyperbola xy=1.


I don't know where to start. I tried graphing it but it doesn't make any sense to me. Thanks for any help!

 Nov 2, 2019

Let A = (a,b), B = (c,d), and C = (e,f).  Use perpendicular bisectors to compute the center of the circumcircle of ABC.  The computations are really easy.

 Nov 5, 2019

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