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In geometry, the '''orthopole''' of a system consisting of a triangle ''ABC'' and a line ''ℓ'' in the same plane is a point determined as follows.<ref>{{cite web | title=MathWorld: Orthopole| url=http://mathworld.wolfram.com/Orthopole.html}}</ref> Let {{nowrap|''A'' {{prime}}, ''B'' {{prime}}, ''C'' {{prime}}}} be the feet of perpendiculars dropped on ''ℓ'' from {{nowrap|''A'', ''B'', ''C''}} respectively. Let {{nowrap|''A'' {{prime}}{{prime}}, ''B'' {{prime}}{{prime}}, ''C'' {{prime}}{{prime}}}} be the feet of perpendiculars dropped from {{nowrap|''A'' {{prime}}, ''B'' {{prime}}, ''C'' {{prime}}}} to the sides opposite {{nowrap|''A'', ''B'', ''C''}} (respectively) or to those sides' extensions. Then the three lines {{nowrap|''A'' {{prime}} ''A'' {{prime}}{{prime}}, ''B'' {{prime}} ''B'' {{prime}}{{prime}}, ''C'' {{prime}} ''C'' {{prime}}{{prime}},}} are concurrent.<ref>{{cite journal |last1=Goormaghtigh |first1=R. |title=The Orthopole |journal=Tohoku Mathematical Journal |series=First Series |date=1926 |volume=27 |pages=77–125 |url=https://www.jstage.jst.go.jp/article/tmj1911/27/0/27_0_77/_article/-char/ja/ }}</ref> The point at which they concur is the orthopole.
Due to their many properties,<ref>{{cite web|url=https://www.geogebra.org/m/CKKH9ZZA|title=The Orthopole|date=21 January 2017|publisher=}}</ref> orthopoles have been the subject of a large literature.<ref>{{cite journal |last1=Ramler |first1=O. J. |title=The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle |journal=The American Mathematical Monthly |date=1930 |volume=37 |issue=3 |pages=130–136 |doi=10.2307/2299415 |jstor=2299415 }}</ref> Some key topics are determination of the lines having a given orthopole<ref>{{cite journal |last1=Karl |first1=Mary Cordia |title=The Projective Theory of Orthopoles |journal=The American Mathematical Monthly |date=1932 |volume=39 |issue=6 |pages=327–338 |doi=10.2307/2300757 |jstor=2300757 }}</ref> and orthopolar circles.<ref>{{cite journal |last1=Goormaghtigh |first1=R. |title=1936. The orthopole |journal=The Mathematical Gazette |date=December 1946 |volume=30 |issue=292 |pages=293 |doi=10.2307/3610737 |jstor=3610737 |s2cid=185932136 }}</ref>
==Literature== * Orthopole=Ортополюс. In Russian
==References== {{Reflist}}
Category:Points defined for a triangle