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Journal for Geometry and Graphics 24 (2020), No. 1, 009--028
Copyright Heldermann Verlag 2020

Constructions in the Locus of Isogonal Conjugates for a Quadrilateral

Daniel Hu
Los Altos High School, 201 Almond Avenue, Los Altos, CA 94022, U.S.A.

Given fixed distinct points A, B, C, D, we examine properties of and constructions involving the cubic plane curve locus of points X for which (XA, XC), (XB, XD) are isogonal. We establish a necessary and sufficient condition for all cubics Cu in real plane R2 such that there exist corresponding points A, B, C, D in Cu for which Cu is this exact locus.

Keywords: isogonal cubic, focal cubic, isogonal conjugate, excellent point, spiral center of quadrangle, Miquel point.

MSC: 51N20; 51N35, 51N15.

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