|
|
If perpendiculars 
,
, and 
are dropped on any line 
from the vertices of a triangle 
, then the perpendiculars to the
opposite sides from their perpendicular feet 
, 
, and 
are concurrent at a point
called the orthopole. The orthopole
of a line lies on the Simson line which is perpendicular
to it (Honsberger 1995, p. 130). If a line crosses the circumcircle
of a triangle, the Simson lines of the points of intersection
meet at the orthopole of the line. Also, the orthopole of a line through the circumcenter 
of a triangle 
lies on that triangle's nine-point
circle (Honsberger 1995, p. 127).
If the line 
is displaced parallel to itself, the orthopole moves
along a line perpendicular to 
a distance equal to the displacement. If 
is the Simson line of a point
, then 
is called the Simson line pole
of 
(Honsberger 1995, p. 128).
The orthopole of a line 
is equivalent to the orthojoin of Kimberling center 
.
The following table summarized the orthopoles for some named central lines.
| line | Kimberling | orthopole | Kimberling |
| antiorthic axis |  |  | |
| Brocard axis |  | center of the Kiepert hyperbola |  |
| de Longchamps line |  |  | |
| Euler line |  | center of the Jerabek hyperbola |  |
| Fermat axis |  | * | |
| Gergonne line |  |  | |
| Lemoine axis |  |  | |
| line at infinity |  | * | |
| Nagel line |  | * | |
| orthic axis |  |  | |
| Soddy line |  |  |
