__notoc__ In

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 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 A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...

is constructed by reflecting the lines about the

angle bisectors In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes throug ...

of respectively. These three reflected lines

concur In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...

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 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 ob ...

is (by definition) the

symmedian point In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the co ...

. 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 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 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 t ...

, 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 In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...

, and the inverse of each in is . As isogonal conjugation is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

, 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 In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld ...

; 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 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 descript ...

, 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, cal ...

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 polyg ...

in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the

line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The l ...

. 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 mat ...

) 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