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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 definition applies only to points not on a sideline of triangle .) 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 (by definition) 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 <math>X=x:y:z</math> is a point not on a sideline of triangle , then its isogonal conjugate is <math>\tfrac{1}{x} : \tfrac{1}{y} : \tfrac{1}{z}.</math> For this reason, the isogonal conjugate of is sometimes denoted by . The set of triangle centers under the trilinear product, defined by

: <math>(p:q:r)*(u:v:w) = pu:qv:rw,</math>

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 cubics (e.g., Thompson cubic, Darboux cubic, Neuberg cubic) are self-isogonal-conjugate, in the sense that if is on the cubic, then is also on the cubic.

Another construction for the isogonal conjugate of a point

thumb|A second definition of isogonal conjugate

For a given point in the plane of triangle , let the reflections of in the sidelines be . Then the center of the circle is the isogonal conjugate of .

See also

  • Isotomic conjugate
  • Central line (geometry)
  • Triangle center

References

  • Interactive Java Applet illustrating isogonal conjugate and its properties
  • MathWorld
  • Pedal Triangle and Isogonal Conjugacy