If perpendiculars A', B', and C' are dropped on any line L from the vertices of a triangle Δ A B C, then the perpendiculars to the opposite sides from their perpendicular feet A'', B'', and C'' are concurrent at a point P called the orthopole. The orthopole of a line lies on the Simson line which is perpendicular to it. 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 O of a triangle Δ A B C lies on that triangle's nine-point circle.

Lemoyne's theorem | nine-point circle | orthopolar line | Rigby points | Simson line | Simson line pole