# Isogonal

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

## Contents

### 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].