- Steve Phelps
If perpendiculars are dropped to any line from the vertices of a triangle, then the perpendiculars to the opposite sides from their perpendicular feet are concurrent at a point called the orthopole.
The orthopole of a line through the circumcenter of a triangle lies on the nine-point circle of the triangle. (click on the reset icon in the upper right corner to erase the traces)
The orthopole of a line lies on the Simson Line that is perpendicular to the line.
If the line crosses the circumcircle of a triangle, the Simson Lines of the two points of intersection will intersect at the orthopole of the line.
Given a line, choose any three of the vertices of a quadrilateral. The orthopoles of the give line with respect to the triangles are collinear. This line is the orthopolar line of the given line with respect to the quadrilateral.