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X(2), X(6), X(13), X(14), X(15), X(16), X(111), X(368), X(524), E(458)=X(368)* vertices of the second and fourth Brocard triangles foci F1, F2 of the inscribed Steiner ellipse. foci of the K-ellipse (inellipse with center K when the triangle ABC is acute angle) other points below |
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K018 is a circular isogonal focal nK with root X(523) and singular focus X(111). The real asymptote is parallel to GK. It is also the orthopivotal cubic O(X6) and Z+(L) with L = X(3)X(6) in TCCT p.241. See also Z+(O) = CL025 and CL034. K018 is spK(X524, X2) in CL055. Locus properties
Miscellaneous properties |
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The polar conic of X(110) decomposes into the Brocard axis and the trilinear polar of the root X(523). It follows that the six tangents at K, X(15), X(16) and the three points on the sidelines of ABC concur at X(110). The common tangential of X(13) and X(15) is T1 and the common tangential of X(14) and X(16) is T2. These two points are collinear with the singular focus X(111). The third points on X(13)X(15) and X(14)X(16) are N1, N2 respectively also collinear with X(111). The midpoint of N1N2 is X(230) on GK. The lines T1N2 and T2N1 are parallel to the asymptote. The tangents at X(13) and X(14) met at the reflection of X(110) about G. The polar conic of X(111) is the circle through G, O, K, X(691) and naturally X(111). The tangent at X(111) to this circle meets the asymptote at X on the cubic. Recall that this asymptote is the homothetic of the axis GK under h(X111, 2). The polar conic of the point at infinity X(524) – that of the line GK – is a rectangular hyperbola passing through X(597), having one asymptote perpendicular at X(111) to GK and the other that of the cubic. The bisectors of the lines X(111)X(524) and X(111)X contain X(2469), X(2470) and the centers of anallagmaty E1, E2 which obviously also lie on the previous rectangular hyperbola. The cubic is then invariant under the two inversions with pole Ei which swap X(111) and Ej. K018 is also invariant under the Psi transformation which is the product of the reflection about one axis of the Steiner inellipse and the inversion with circle that of diameter F1F2, the foci of the ellipse. See also "Orthocorrespondence and Orthopivotal Cubics", §5 and K022. Psi is the involution that swaps any point of the plane to the center of its polar conic in the McCay or Kjp cubics. In particular, Psi swaps A, B, C and the vertices of the second Brocard triangle. |
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K018 contains the equi-Brocard center X(368), its isogonal conjugate X(368)* and its Psi image. Recall that X(368) also lies on the Wallace hyperbola (the anticomplement of the Kiepert hyperbola) and on the three equibrocardian focal cubics K083. |
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