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Given a pole W of isoconjugation which is not the Lemoine point K, there is one and only one focal (circular) nK with pole W. Let L be the Wisoconjugate of the circumcircle i.e. the trilinear polar of the isotomic conjugate of the isogonal conjugate of W. Denote by :


R is the root of the focal cubic and the trilinear polar of R meets the sidelines of ABC at U, V, W on the cubic. F is the singular focus and X the intersection of the cubic with its real asymptote, the homothetic of L under h(F,2). The circle with diameter FX passes through P and S. Thus, the lines PF, PX and SF, SX are perpendicular. The cubic passes through the orthogonal projection E of F on the line SP and D intersection of the lines FS, PS*. D and E are isoconjugate. The tangents at P, S, F and the real asymptote concur at X hence the polar conic of X is the hyperbola passing through P, S, F, S* and X. The polar conic of F is the circle passing through F, centered on PS and orthogonal to the circle with diameter PS. In other words, its center is the intersection of PS with the perpendicular at F to FX. 

Locus properties :


Other properties :


The table gives a short selection of nonisogonal focal nKs (the red point is the singular focus). 


