The Simson line is the line containing the feet , , and of the perpendiculars from an arbitrary point on the circumcircle of a triangle to the sides or their extensions of the triangle. This line was attributed to Simson by Poncelet, but is now frequently known as the Wallace-Simson line since it does not actually appear in any work of Simson (Johnson 1929, p. 137; Coxeter and Greitzer 1967, p. 41; de Guzmán 1999). The inverse statement to that given above, namely that the locus of all points in the plane of a triangle such that the feet of perpendiculars from the three sides of the triangle is collinear is given by the circumcircle of , is sometimes called the Wallace-Simson theorem (de Guzmán 1999).
The trilinear equation of the Simson line for a point lying on the circumcircle, i.e., satisfying
is
(P. Moses, pers. comm., Jan. 27, 2005).
The Simson line bisects the line , where is the orthocenter (Honsberger 1995, p. 46). Moreover, the midpoint of lies on the nine-point circle (Honsberger 1995, pp. 46-47). The Simson lines of two opposite point on the circumcenter of a triangle are perpendicular and meet on the nine-point circle.
The angle between the Simson lines of two points and is half the angle of the arc . The Simson line of any polygon vertex is the altitude through that polygon vertex. The Simson line of a point opposite a polygon vertex is the corresponding side. If is the Simson line of a point of the circumcircle, then the triangles and are directly similar.
The envelope of the Simson lines of a triangle is a deltoid (Butchart 1939; Wells 1991, pp. 155 and 230). The area of the deltoid is half the area of the circumcircle (Wells 1991, p. 230), and the first Morley triangle of the starting triangle has the same orientation as the deltoid. Each side of the triangle is tangent to the deltoid at a point whose distance from the midpoint of the side equals the chord of the nine-point circle cut off by that side (Wells 1991, p. 231). If a line is the Simson line of a point on the circumcircle of a triangle, then is called the Simson line pole of (Honsberger 1995, p. 128).
The altitudes of a reference triangle are Simson lines whose Simson line poles are the vertices of the reference triangle. Furthermore, the sides of the reference triangle are also Simson lines whose Simson line poles are the reflections of the vertices of the reference triangle about its circumcenter. Note also that the nontrivial perpendicular feet from these reflective vertices intersect the sides of the reference triangle at points that are the tangents to the Steiner deltoid.