Isogonal conjugate

The definition of isogonal conjugate points with respect to a triangle arises from the following set:

It is a triangle ABC and P is a point which does not lie on the sides of the triangle. We mirror the lines AP, BP and CP to the bisectors of the triangle angles α, β and γ. Then the mirror images intersect at a new point.

This new point is called the isogonal conjugate point with respect to P of the triangle ABC ( the clause "with respect to the triangle ABC ," you can usually away when no confusion is to be feared, so if only one triangle is in play ). We refer to the P isogonal conjugate point Q, the Q to isogonal conjugate point is in turn the point P. Thus, the points P and Q as each other isogonal conjugate points, respectively.

It should be noted that the term "point" is to expand here on the projective plane. Namely, it can happen that the mirror images of the straight line AP, BP and CP to the α angle bisector of the triangular angle, β and γ are parallel to each other; then you say that they are in an infinitely distant point (in short: Remote point) cut. The point Q is thus in this case a remote point. ( One can show that this case - that Q is a remote point -. Exactly occurs when the point P lies on the circumcircle of triangle ABC)

Examples

  • There are exactly four points that are conjugated isogonal with respect to the triangle ABC to himself, the incenter and the three Ankreismittelpunkte.
  • The circumcenter is isogonal conjugate to the orthocenter.
  • The focus is isogonal conjugated to Lemoinepunkt.
  • The first and second Brocard points are isogonal conjugate to each other.
  • The cornerstones of the anti medial triangle and the corresponding corners of the triangle are tangent pairs, conjugated isogonal.

Properties

  • Has a point, the trilinear coordinates, the trilinear coordinates of the isogonal conjugate point are given by

.

  • Has a point, the barycentric coordinates, the barycentric coordinates of the isogonal conjugate point are given by. Here are the names, and for the side lengths of the triangle.
  • The Fußpunktdreiecke two isogonal conjugate points have the same radius.
  • By forming the points of a line to the corresponding isogonal conjugate points from, then a conic that passes through the vertices of the given triangle. The type of this conic section depends on how the given line and the perimeter of the triangle are: Cuts the line the perimeter, there is a hyperbola. If the straight line is a tangent of the circumference, the result is a parabola. If the straight line no common points with the radius, has obtained an ellipse.

Equivalent definition

P is a point of the triangle ABC, which is not located on the sides thereof. Reflects one P to the sides BC, AC and AB or their extensions, so arise the three image points Pa, Pb and Pc. The circumcenter of triangle Q PaPbPc then called the isogonal conjugate point with respect to P of triangle ABC.

Extension of the concept

One can carry out the construction of the isogonal conjugate point Q for points P on the sides of the triangle ABC, but not for its vertices. The resulting pixels are then always the opposite corners of the triangle ABC. This is the mapping which assigns to each point its isogonal conjugate point, no longer injective and one can no longer speak of two mutually isogonal conjugate points P and Q.

Inscribed circle Isotomic conjugate Wolfram Research
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