Literally "same angle" . There are several concepts in mathematics involving isogonality.
A trajectory that meets a given family of curves at a constant angle. See Isogonal trajectory.
A circle is said to be isogonal with respect to two other circles if it makes the same angle with these two, [a1].
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
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.
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].
|[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)|
Isogonal. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Isogonal&oldid=39653