The Simson line is the line containing the perpendicular 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 perpendicular
feet on the three sides of the triangle
are collinear is given by the circumcircle
of 
, is sometimes called the Wallace-Simson
theorem (de Guzmán 1999).
The equation of the Simson line in trilinear coordinates for a point 
lying on the circumcircle,
i.e., satisfying
 |
(1)
|
is
 |
(2)
|
(P. Moses, pers. comm., Jan. 27, 2005).
This equation applies when 
is not a polygon vertex of the reference
triangle. At a polygon vertex, all three coefficients vanish, and the Simson line is the corresponding
altitude.
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 points
on the circumcircle 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 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 of tangency with the Steiner deltoid.
Carnot's equal-angle extension of the Simson line theorem replaces the perpendiculars by lines
making a common directed angle with the cyclically
oriented sides. For 
on the circumcircle of triangle 
, let 
, 
,
and 
be the finite intersections
of those lines with the extended sides 
, 
, and 
, respectively, with 
, 
,
and 
defined. If
 |
(3)
|
then 
, 
, and 
are collinear. The choice of
a right angle recovers the Simson line (Tran Minh Ngoc
2018, Theorem 3.6).
