Orthopole Explained

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.[1] Let be the feet of perpendiculars dropped on ℓ from respectively. Let be the feet of perpendiculars dropped from to the sides opposite (respectively) or to those sides' extensions. Then the three lines are concurrent.[2] The point at which they concur is the orthopole.

Due to their many properties,[3] orthopoles have been the subject of a large literature.[4] Some key topics are determination of the lines having a given orthopole[5] and orthopolar circles.[6]

Literature

Notes and References

  1. Web site: MathWorld: Orthopole.
  2. Goormaghtigh . R. . The Orthopole . Tohoku Mathematical Journal . First Series . 1926 . 27 . 77–125 .
  3. Web site: The Orthopole. 21 January 2017.
  4. Ramler . O. J. . The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle . The American Mathematical Monthly . 1930 . 37 . 3 . 130–136 . 10.2307/2299415 . 2299415 .
  5. Karl . Mary Cordia . The Projective Theory of Orthopoles . The American Mathematical Monthly . 1932 . 39 . 6 . 327–338 . 10.2307/2300757 . 2300757 .
  6. Goormaghtigh . R. . 1936. The orthopole . The Mathematical Gazette . December 1946 . 30 . 292 . 293 . 10.2307/3610737 . 3610737 . 185932136 .

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