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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 u x + v y + w z = 0. {\displaystyle ux+vy+wz=0.} This line meets the circumcircle of ABC in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.
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.
If does not lie on the circumcircle then its isogonal conjugate yields the same pedal circle, that is the six points ,, and ,, lie on the same circle. Moreover, the midpoint of the line segment P Q {\displaystyle PQ} is the center of that pedal circle.
The hyperbola is the isogonal conjugate of , the line joining the circumcenter and the incenter. [3] This fact leads to a few interesting properties. Specifically all the points lying on the line O I {\displaystyle OI} have their isogonal conjugates lying on the hyperbola.
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 isogonal conjugate of P is sometimes denoted by P*. The isogonal conjugate of P* is P. A self-isogonal cubic is a triangle cubic that is invariant under isogonal conjugation. A pivotal isogonal cubic is a cubic in which points P lying on the cubic and their isogonal conjugates are collinear with a fixed point Q known as the pivot point of ...
Define D' as the isogonal conjugate of D. It is easy to see that the reflection of CD about the bisector is the line through C parallel to AB. The same is true for BD, and so, ABD'C is a parallelogram. AD' is clearly the median, because a parallelogram's diagonals bisect each other, and AD is its reflection about the bisector. third proof.