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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 |