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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 .
Interactive Java Applet illustrating isogonal conjugate and its properties
Pedal Triangle and Isogonal Conjugacy
Triangle geometry
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the isogonal conjugate of a point with respect to a triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colli ...
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
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
is (by definition) the symmedian point . The isogonal conjugates of the Fermat point
In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...
s are the isodynamic point
In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...
s and vice versa. The Brocard points are isogonal conjugates of each other.
In trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...
, 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
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, or hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, c ...
according as the line intersects the circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every pol ...
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
In Euclidean geometry, the Neuberg cubic is a special cubic plane curve associated with a reference triangle with several remarkable properties. It is named after Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926), a Luxembourger m ...
) 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
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
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For examp ...
References
External links
{{commonscat, Isogonal ConjugatesInteractive Java Applet illustrating isogonal conjugate and its properties
Pedal Triangle and Isogonal Conjugacy
Triangle geometry