__notoc__ In geometry, the isogonal conjugate of a

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * 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 In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are kn ...

. The isogonal conjugate of a point is sometimes denoted by . The isogonal conjugate of is . The isogonal conjugate of the

incentre In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...

is itself. The isogonal conjugate of the

orthocentre In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

is the

circumcentre 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 polyg ...

. 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 In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

are isogonal conjugates of each other. In trilinear coordinates, if

X=x:y:z

is a point not on a sideline of triangle , then its isogonal conjugate is

\tfrac : \tfrac : \tfrac.

For this reason, the isogonal conjugate of is sometimes denoted by . The

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of triangle centers under the trilinear product, defined by :

(p:q:r)*(u:v:w) = pu:qv:rw,

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 In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, 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

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 .

References

External links

{{commonscat, Isogonal Conjugates
Interactive Java Applet illustrating isogonal conjugate and its properties

Pedal Triangle and Isogonal Conjugacy
Triangle geometry

Another construction for the isogonal conjugate of a point

See also

References

External links