The isogonal conjugate of a point in the plane of the triangle is constructed by reflecting
the lines ,
,
and
about the angle bisectors at , , and . The three reflected lines then concur
at the isogonal conjugate (Honsberger 1995, pp. 55-56). In older literature,
isogonal conjugate points are also known as counter points (Gallatly 1913), Gegenpunkte
(Gallatly 1913), and focal pairs (Morley 1954).
Isogonal conjugation maps the interior of a triangle onto itself. This mapping transforms lines into circumconics.
The type of conic section is determined by whether
the line
meets the circumcircle ,
The isogonal conjugate of a point on the circumcircle is a point at infinity (and conversely). The
sides of the pedal triangle of a point are perpendicular
to the connectors of the corresponding polygon vertices
with the isogonal conjugate. The isogonal conjugate of a set of points is the locus of their isogonal conjugate points.
The product of isotomic and isogonal conjugation is a collineation which transforms the sides of
a triangle to themselves (Vandeghen 1965).