# Isogonal

Literally "same angle" . There are several concepts in mathematics involving isogonality.

### Isogonal trajectory.

A trajectory that meets a given family of curves at a constant angle. See Isogonal trajectory.

### Isogonal mapping.

A (differentiable) mapping that preserves angles. For instance, the stereographic projection of cartography has this property [a2]. See also Conformal mapping; Anti-conformal mapping.

### Isogonal circles.

A circle is said to be isogonal with respect to two other circles if it makes the same angle with these two, [a1].

### Isogonal line.

Given a triangle and a line from one of the vertices, say from , to the opposite side. The corresponding isogonal line is obtained by reflecting with respect to the bisectrix in .

If the lines , and are concurrent (i.e. pass through a single point , i.e. are Cevian lines), then so are the isogonal lines , , . This follows fairly directly from the Ceva theorem. The point is called the isogonal conjugate point. If the barycentric coordinates of (often called trilinear coordinates in this setting) are , then those of are

Figure: i130080a

Another notion in rather the same spirit is that of the isotomic line to , which is the line such that . Again it is true that if , , are concurrent, then so are , , . This follows directly from the Ceva theorem.

Figure: i130080b

The point is called the isotomic conjugate point. The barycentric coordinates of are , where , , are the lengths of the sides of the triangle. The Gergonne point is the isotomic conjugate of the Nagel point.

The involutions and , i.e. isogonal conjugation and isotomic conjugation, are better regarded as involutions of the projective plane , [a4].

#### References

[a1] | M. Berger, "Geometry" , I , Springer (1987) pp. 327 |

[a2] | D. Hilbert, S. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) pp. 249 |

[a3] | R.A. Johnson, "Modern geometry" , Houghton–Mifflin (1929) |

[a4] | R.H. Eddy, J.B. Wilker, "Plane mappings of isogonal-isotomic type" Soochow J. Math. , 18 : 2 (1992) pp. 135–158 |

[a5] | N. Altshiller–Court, "College geometry" , Barnes & Noble (1952) |

[a6] | H.S.M. Coxeter, "The real projective plane" , Springer (1993) pp. 197–199 (Edition: Third) |

[a7] | F. Bachmann, "Aufbau der Geometrie aus dem Spiegelungsbegriff" , Springer (1973) (Edition: Second) |

**How to Cite This Entry:**

Isogonal.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Isogonal&oldid=12429