Isogonal conjugate
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Isogonal conjugate transformation over the points inside the triangle.
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In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of. This is a direct result of the trigonometric form of Ceva's theorem.
The isogonal conjugate of a point is sometimes denoted by. The isogonal conjugate of is.
The isogonal conjugate of the incentre is itself. The isogonal conjugate of the orthocentre is the circumcentre. The isogonal conjugate of the centroid is the symmedian point. 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, then its isogonal conjugate is For this reason, the isogonal conjugate of is sometimes denoted by. The set of triangle centers under the trilinear product, defined by
is a commutative group, and the inverse of each in is.
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 Cubic [plane curve|cubics] are self-isogonal-conjugate, in the sense that if is on the cubic, then is also on the cubic.