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As isogonal conjugation is a function, it makes sense to speak of the isogonal conjugate of sets of points, such as lines and circles. For example, the isogonal conjugate of a line is a circumconic; specifically, an ellipse, parabola, or hyperbola according as the line intersects the circumcircle in 0, 1, or 2 points.
The isogonal conjugate of each point X on the circumconic, other than A, B, C, is a point on the line + + = This line meets the circumcircle of ABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
Specifically all the points lying on the line have their isogonal conjugates lying on the hyperbola. The Nagel point lies on the curve since its isogonal conjugate is the point of concurrency of the lines joining the vertices and the opposite Mixtilinear incircle touchpoints, also the in-similitude of the incircle and the circumcircle.
The isogonal conjugate of the centroid X 2 is the symmedian point X 6 (also denoted by K) having trilinear coordinates a : b : c. So the Lemoine axis of ABC is the trilinear polar of the symmedian point of ABC. The tangential triangle of ABC is the triangle T A T B T C formed by the tangents to the circumcircle of ABC at its vertices.
The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this: (,,,,,) =. The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of ABC. A construction of X* follows.
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If P and Q are isogonal conjugates with respect to ABC, then the Ceva product of their complements lies on the Kiepert hyperbola. Theorem 9.1. The Yff center of congruence is the internal center of similitude of the incircle and the circumcircle with respect to the pedal triangle of the incenter.