Orthopole

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:

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. The point at which they concur is the orthopole.

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

Literature

Orthopole=Ортополюс. In Russian

References

↑ "MathWorld: Orthopole".
↑ Goormaghtigh, R. (1926). "The Orthopole". Tohoku Mathematical Journal. First Series. 27: 77–125.
↑ "The Orthopole". 21 January 2017.
↑ 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.
↑ Karl, Mary Cordia (1932). "The Projective Theory of Orthopoles". The American Mathematical Monthly. 39 (6): 327–338. doi:10.2307/2300757. JSTOR 2300757.
↑ Goormaghtigh, R. (December 1946). "1936. The orthopole". The Mathematical Gazette. 30 (292): 293. doi:10.2307/3610737. JSTOR 3610737. S2CID 185932136.

[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.