Orthopole

  Triangle △ABC
  Line ℓ
  Perpendiculars AA', BB', CC'
  Perpendiculars A'A", B'B", C'C" (intersect at orthopole H)

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 A', B', C' be the feet of perpendiculars dropped on ℓ from A, B, C respectively.
  • Let A", B", C" be the feet of perpendiculars dropped from A', B', C' to the sides opposite A, B, C (respectively) or to those sides' extensions.
  • Then the three lines A'A", B'B", C'C", 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

References

  1. ^ "MathWorld: Orthopole".
  2. ^ Goormaghtigh, R. (1926). "The Orthopole". Tohoku Mathematical Journal. First Series. 27: 77–125.
  3. ^ "The Orthopole". 21 January 2017.
  4. ^ Ramler, O. J. (1930). "The Orthopole Loci of Some One-Parameter Systems of Lines Referred to a Fixed Triangle". The American Mathematical Monthly. 37 (3): 130–136. doi:10.2307/2299415. JSTOR 2299415.
  5. ^ Karl, Mary Cordia (1932). "The Projective Theory of Orthopoles". The American Mathematical Monthly. 39 (6): 327–338. doi:10.2307/2300757. JSTOR 2300757.
  6. ^ Goormaghtigh, R. (December 1946). "1936. The orthopole". The Mathematical Gazette. 30 (292): 293. doi:10.2307/3610737. JSTOR 3610737. S2CID 185932136.

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