Isogonal Conjugate

The isogonal conjugate $X^{-1}$ of a point $X$ in the plane of the Triangle $\Delta ABC$ is constructed by reflecting the lines $AX$, $BX$, and $CX$ about the Angle Bisectors at $A$, $B$, and $C$. The three reflected lines Concur at the isogonal conjugate. The Trilinear Coordinates of the isogonal conjugate of the point with coordinates

$$\alpha:\beta:\gamma$$

are

$$\alpha^{-1}:\beta^{-1}:\gamma^{-1}.$$

Isogonal conjugation maps the interior of a Triangle onto itself. This mapping transforms lines onto Conic Sections that Circumscribe the Triangle. The type of Conic Section is determined by whether the line $d$ meets the Circumcircle $C'$,

1. If $d$ does not intersect $C'$, the isogonal transform is an Ellipse;

2. If $d$ is tangent to $C'$, the transform is a Parabola;

3. If $d$ cuts $C'$, the transform is a Hyperbola, which is a Rectangular Hyperbola if the line passes through the Circumcenter

(Casey 1893, Vandeghen 1965).

The isogonal conjugate of a point on the Circumcircle is a Point at Infinity (and conversely). The sides of the Pedal Triangle of a point are Perpendicular to the connectors of the corresponding Vertices with the isogonal conjugate. The isogonal conjugate of a set of points is the Locus of their isogonal conjugate points.

The product of Isotomic and isogonal conjugation is a Collineation which transforms the sides of a Triangle to themselves (Vandeghen 1965).

See also Antipedal Triangle, Collineation, Isogonal Line, Isotomic Conjugate Point, Line at Infinity, Symmedian Line

References

Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions with Numerous Examples, 2nd rev. enl. ed. Dublin: Hodges, Figgis, & Co., 1893.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 153-158, 1929.

Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.