Isogonal conjugate
In geometry, the isogonal conjugate of a point P with respect to a triangle △ABC is constructed by reflecting the lines PA, PB, PC about the angle bisectors of A, B, C respectively.
These three reflected lines concur at the isogonal conjugate of P. (This definition applies only to points not on a sideline of triangle △ABC.)
This is a direct result of the trigonometric form of Ceva's theorem.
The set S of triangle centers under the trilinear product, defined by is a commutative group, and the inverse of each X in S is X –1.
For a given point P in the plane of triangle △ABC, let the reflections of P in the sidelines BC, CA, AB be Pa, Pb, Pc.
Lines from each vertex to
P
Lines to
P
reflected about the angle bisectors (concur at
P*
, the
isogonal conjugate
of
P
)
