IMO 1969 LL BEL2

Origin: BEL

Problem

(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.