Focal Midpoints, Rectangular Hyperbolas, and Orthocentre Reconstruction

The midpoints of equal-product focal pairs for a nonrectangular cyclic quadruple form a rectangular hyperbola with one excluded point. An orthocentre closure property yields unique focal reconstruction from three admissible midpoints. For a quadruple on a triangle's nine-point circle, the excluded point is the midpoint of the circumcentre and the fourth anchor.