Find the equations of regular hyperbolas passing through the points and Prove that all such hyperbolas pass through the orthocenter of the triangle Find the locus of the centers of these hyperbolas. Check whether this locus coincides with the nine-point circle of the triangle
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$(BEL 2) (a)$ 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 $ABC.$
Izvor: Međunarodna matematička olimpijada, shortlist 1969
Školjka 
Find the equations of regular hyperbolas passing through the points 
and 
Prove that all such hyperbolas pass through the orthocenter 
of the triangle 
Find the locus of the centers of these hyperbolas.
Check whether this locus coincides with the nine-point circle of the triangle