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X(1), X(2), X(3), X(4), X(6) 

K383 is the only (undecomposed) isogonal cK with node I passing through I, G, O, H, K. Hence it has nine known common points with the Thomson cubic. Its root is X(14543), the intersection of the lines X(2)X(1762) and X(7)X(1781), a point with first barycentric a^2(b+c+2a)/(bc)+(b^2c^2). The pivotal conic is inscribed in the excentral triangle. It is tangent to the line GK and to the Euler line which gives its construction and the construction of the cubic. The tangent at O is the Brocard axis and the tangent at H is the line HK. The polar conic of K contains I, O, H, K, X(71), X(1474), X(1780). See a generalization at Table 30. K383 is a member of the class CL061. 
