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Equations, hyperbolas, Locus and nine point circle.

Source:

September 29, 2010
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Problem Statement

(BEL2)(a)(BEL 2) (a) Find the equations of regular hyperbolas passing through the points A(α,0),B(β,0),A(\alpha, 0), B(\beta, 0), and C(0,γ).C(0, \gamma). (b)(b) Prove that all such hyperbolas pass through the orthocenter HH of the triangle ABC.ABC. (c)(c) Find the locus of the centers of these hyperbolas. (d)(d) Check whether this locus coincides with the nine-point circle of the triangle ABC.ABC.
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