Question 2
(BEL 2)
(a) Find the equations of regular hyperbolas passing through the points $A(\alpha,0)$, $B(\beta,0)$, and $C(0,\gamma)$.
(b) Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC$.
(c) Find the locus of the centers of these hyperbolas.
(d) Check whether this locus coincides with the nine-point circle of the triangle $ABC$.
(a) Find the equations of regular hyperbolas passing through the points $A(\alpha,0)$, $B(\beta,0)$, and $C(0,\gamma)$.
(b) Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC$.
(c) Find the locus of the centers of these hyperbolas.
(d) Check whether this locus coincides with the nine-point circle of the triangle $ABC$.