Igor Minevich and Patrick Morton, A quadrilateral half-turn theorem,
Forum Geometricorum, 16 (2016) 133--139.

Abstract. If ABC is a given triangle in the plane, P is any point not on the extended sides of ABC or its anticomplementary triangle, Q is the complement of the isotomic conjugate of P with respect to ABC, DEF is the cevian triangle of P, and D_0 and A_0 are the midpoints of segments BC and EF, respectively, a synthetic proof is given for the fact that the complete quadrilateral defined by the lines AP, AQ, D_0Q, D_0A_0 is perspective by a Euclidean half-turn to the similarly defined complete quadrilateral for the isotomic conjugate P' of P. This fact is used to define and prove the existence of a generalized circumcenter and generalized orthocenter for any such point P.

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