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X(3), X(4), X(30), X(110) four foci of the MacBeath inconic : X(3), X(4) and two imaginary other points below |
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K799 is a focal cubic related to K463, K725. Its singular focus is X(110) whose polar conic is the circle passing through X(3), X(4), X(110). It is the locus of contacts of tangents drawn through X(110) to the circles passing through X(3), X(4). K799 passes through : • the common points Q1, Q2, Q3 of the Orthocubic K006 and the circumcircle (apart A, B, C). • the intersections R1, R2, R3 of the perpendicular bisector of X(4)X(110) with the sidelines of Q1Q2Q3. • six (not always real) points (labelled Ae, Ai, etc) on the perpendicular bisectors of ABC, two by two symmetric about the sidelines of ABC. K799 is a member of the pencil generated by the Neuberg cubic K001 and the union of the line at infinity (counted twice) with the Euler line. See K725 for further details and a generalization. |