Isogonal Regions

This page shows how a point relates to its isotomic conjugate. This page is taken from The Triangle Book. Some details came from my own research. Another page shows isotomic regions.

The isogonal conjugate: Each sense of center leads to a sense of deviation from center. For points the obvious center is the midpoint; for lines it is the angle bisector. For lines the deviation from center is the angle from the bisector. Equal deviations are called isogonal.

Consider a point P in the plane of the triangle. The intersection of the Cevian lines through P with the triangle edges are called the Cevian traces of P. The isogonal conjugate of P is formed by the concurrence of the lines formed from the reflections of the lines over the bisector from its vertex.

Figure: Along each edge, the traces of P and its isogonal conjugate gP are equally spaced to the midpoint. At vertex A both bisectors are shown, showing that reflection across either bisector gives the same result.