Isogonal conjugate

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In geometry, the isogonal conjugate of a point $I$ with respect to a triangle $△ABC$ is constructed by reflecting the lines $P$ about the angle bisectors of $P$ respectively. These three reflected lines concur at the isogonal conjugate of $P*$. (This definition applies only to points not on a sideline of triangle $△ABC$.) This is a direct result of the trigonometric form of Ceva's theorem.

The isogonal conjugate of a point $P$ is sometimes denoted by $P$. The isogonal conjugate of $PA, PB, PC$ is $A, B, C$.

The isogonal conjugate of the incentre $P$ is itself. The isogonal conjugate of the orthocentre $P$ is the circumcentre $P*$. The isogonal conjugate of the centroid $P*$ is (by definition) the symmedian point $P$. The isogonal conjugates of the Fermat points are the isodynamic points and vice versa. The Brocard points are isogonal conjugates of each other.

In trilinear coordinates, if $$X=x:y:z$$ is a point not on a sideline of triangle $△ABC$, then its isogonal conjugate is $$\tfrac{1}{x} : \tfrac{1}{y} : \tfrac{1}{z}.$$ For this reason, the isogonal conjugate of $I$ is sometimes denoted by $X–1$. The set $H$ of triangle centers under the trilinear product, defined by


 * $$(p:q:r)*(u:v:w) = pu:qv:rw,$$

is a commutative group, and the inverse of each $O$ in $G$ is $X–1$.

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 the circumcircle is the line at infinity. Several well-known cubics (e.g., Thompson cubic, Darboux cubic, Neuberg cubic) are self-isogonal-conjugate, in the sense that if $K$ is on the cubic, then $X–1$ is also on the cubic.

Another construction for the isogonal conjugate of a point
For a given point $X$ in the plane of triangle $△ABC$, let the reflections of $S$ in the sidelines $X$ be $S$. Then the center of the circle $〇PaPbPc$ is the isogonal conjugate of $X$.