If perpendiculars A^', B^', and C^' are dropped on any line L from the vertices of a triangle DeltaABC, 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 (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 O of a triangle DeltaABC lies on that triangle's nine-point circle (Honsberger 1995, p. 127).

If the line L is displaced parallel to itself, the orthopole moves along a line perpendicular to L a distance equal to the displacement. If L is the Simson line of a point P, then P is called the Simson line pole of L (Honsberger 1995, p. 128).

The orthopole of a line L_i is equivalent to the orthojoin of Kimberling center X_i.

The following table summarized the orthopoles for some named central lines.

See also

Lemoyne's Theorem, Nine-Point Circle, Orthopolar Line, Rigby Points, Simson Line, Simson Line Pole

Explore with Wolfram|Alpha


Bogomolny, A. "Orthopole.", R. "Analytic Treatment of Some Orthopole Theorems." Amer. Math. Monthly 46, 265-269, 1939.Gallatly, W. "The Orthopole." Ch. 6 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, pp. 46-54, 1913.Honsberger, R. "The Orthopole." Ch. 11 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 125-136, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 247, 1929.Ramler, O. J. "The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle." Amer. Math. Monthly 37, 130-136, 1930.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Orthopole." From MathWorld--A Wolfram Web Resource.

Subject classifications