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# Orthopole

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

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

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## References

Bogomolny, A. "Orthopole." http://www.cut-the-knot.org/Curriculum/Geometry/Orthopole.shtml.Goormaghtigh, 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.

Orthopole

## Cite this as:

Weisstein, Eric W. "Orthopole." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Orthopole.html