Deltoid1Through each point P on the circumcircle of a triangle ABC, a line is drawn in the direction obtained by reflecting line AP in the angle bisector at angle A. (You get the same direction if BP is reflected in the B-bisector, and likewise for C.)
The set of all such lines forms, as an envelope, the Steiner deltoid of the excentral triangle and also of the anticomplementary triangle. Its vertices, called cusps, are the vertices of an equilateral triangle; the name deltoid matches the Greek capital delta, shaped like a triangle. The Steiner deltoid of triangle ABC is smaller than the deltoid shown here, which was described as early as 1966, by Peter Yff. For details about the Steiner deltoid, visit MathWorld.
For more, see GEOMETRY IN ACTION, Chapter 6.