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In geometry, the isogonal conjugate of a point P with respect to a triangle △ABC is constructed by reflecting the lines PA, PB, PC about the angle bisectors of A, B, C 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.
Lines from each vertex to P
Lines to P reflected about the angle bisectors (concur at P*, the isogonal conjugate of P)
The isogonal conjugate of a point P is sometimes denoted by P*. The isogonal conjugate of P* is P.
The isogonal conjugate of the incentre I is itself. The isogonal conjugate of the orthocentre H is the circumcentre O. The isogonal conjugate of the centroid G is (by definition) the symmedian point K. 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 is a point not on a sideline of triangle △ABC, then its isogonal conjugate is For this reason, the isogonal conjugate of X is sometimes denoted by X –1. The set S of triangle centers under the trilinear product, defined by
is a commutative group, and the inverse of each X in S 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 X is on the cubic, then X –1 is also on the cubic.
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For a given point P in the plane of triangle △ABC, let the reflections of P in the sidelines BC, CA, AB be Pa, Pb, Pc. Then the center of the circle 〇PaPbPc is the isogonal conjugate of P.[1]
The isogonal conjugate of the incenter of triangle △ABC is the incenter itself.
The isogonal conjugate of the symmedian point is the third centroid, and the isogonal conjugate of the orthocenter is the circumcenter.
Isogonal conjugates of lines are circumconics, and conversely, isotomic conjugates of circumconics are lines. This property holds for Isotomic conjugate as well.
In May 2021, Dao Thanh Oai gave a generalization of the isogonal conjugate as follows:[2] Let △ ABC be a triangle, P a point on its plane and Ω an arbitrary circumconic of △ ABC. Lines AP, BP, CP cut again Ω at A', B', C' respectively, and parallel lines through these points to BC, CA, AB cut Ω again at A", B", C" respectively. Then lines AA", BB", CC" are concurent.
If barycentric coordinates of the center X of Ω are and , then D, the point of intersection of AA", BB", CC" is:
The point D above is called the X-Dao conjugate of P. This conjugate is a generalization of all known kinds of conjugates:[2]
- When Ω is the circumcircle of ABC, the Dao conjugate becomes the isogonal conjugate of P.
- When Ω is the Steiner circumellipse of ABC, the Dao conjugate becomes the isotomic conjugate of P.
- When Ω is the circumconic of ABC with center X = X(1249), Dao conjugate become the Polar conjugate of P.
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- Isotomic conjugate
- Polar conjugate
- Triangle center
- Central line (geometry)
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