Simson Line
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.
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