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Equal parallelians point

From Wikipedia, the free encyclopedia

In geometry, the equal parallelians point[1][2] (also called congruent parallelians point) is a special point associated with a plane triangle. It is a triangle center and it is denoted by X(192) in Clark Kimberling's Encyclopedia of Triangle Centers.[3] There is a reference to this point in one of Peter Yff's notebooks, written in 1961.[1]

Definition

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  Reference triangle ABC
  Line segments of equal length, parallel to the sidelines of ABC

The equal parallelians point of triangle ABC is a point P in the plane of ABC such that the three line segments through P parallel to the sidelines of ABC and having endpoints on these sidelines have equal lengths.[1]

Trilinear coordinates

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The trilinear coordinates of the equal parallelians point of triangle ABC are

Construction for the equal parallelians point

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Construction of the equal parallelians point.
  Reference triangle ABC
  Internal bisectors of ABC (intersect opposite sides at A", B", C")
  Anticomplementary triangle A'B'C' of ABC
  Lines (A'A", B'B", C'C") concurrent at the equal parallelians point

Let A'B'C' be the anticomplementary triangle of triangle ABC. Let the internal bisectors of the angles at the vertices A, B, C of ABC meet the opposite sidelines at A", B", C" respectively. Then the lines A'A", B'B", C'C" concur at the equal parallelians point of ABC.[2]

See also

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References

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  1. ^ a b c Kimberling, Clark. "Equal Parallelians Point". Archived from the original on 16 May 2012. Retrieved 12 June 2012.
  2. ^ a b Weisstein, Eric. "Equal Parallelians Point". MathWorld--A Wolfram Web Resource. Retrieved 12 June 2012.
  3. ^ Kimberling, Clark. "Encyclopedia of Triangle Centers". Archived from the original on 19 April 2012. Retrieved 12 June 2012.