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Deltoid1

Through 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.