__NOTOC__ In

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the isogonal conjugate of a point with respect to a

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

is constructed by reflecting the lines about the

angle bisectors In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''s ...

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 k ...

. 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 bise ...

is itself. The isogonal conjugate of the

orthocentre The orthocenter of a triangle, usually denoted by , is the point where the three (possibly extended) altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute. For a right triangle, the orthocenter coi ...

is the

circumcentre In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

. 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 figure. The same definition extends to any object in n-d ...

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 Inversive geometry, inversion centered at one of these points transforms the given triangle into an equilateral triang ...

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 with sides , where the vertices are labeled in counterclockwise order, ther ...

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

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 commuta ...

, 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-orie ...

, 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 Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

, or

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, 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, called connected component ( ...

according as the line intersects the

circumcircle In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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 affine plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at ...

. 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

{{commons category, 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