A quadrilateral half-turn theorem
Igor Minevich, Patrick Morton
TL;DR
This work addresses defining and understanding generalized triangle centers attached to an arbitrary point $P$ via a synthetic half-turn symmetry. It proves the Quadrilateral Half-turn Theorem, showing that two carefully constructed complete quadrilaterals associated with $P$ and its isotomcomplement are related by a Euclidean half-turn about $N_1$, and uses this to establish the existence of a generalized circumcenter $O$ and a generalized orthocenter $H$ for $P$ through affine maps and the isotomcomplement $Q=K(P')$. It then derives explicit affine and, in barycentric form, coordinate expressions for $O$ and $H$, relating them to notable centers (e.g., Nagel, Gergonne) and revealing deep symmetries between $P$ and its isotomic conjugate. The results provide a robust framework connecting synthetic geometry, affine transformations, and triangle-center theory, with potential applications to generalized center calculations and coordinate descriptions of triangle configurations.
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 $P$.