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Equal parallelians point

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Triangle center

In geometry, the equal parallelians point (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. There is a reference to this point in one of Peter Yff's notebooks, written in 1961.

Definition

  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.

Trilinear coordinates

The trilinear coordinates of the equal parallelians point of triangle △ABC are b c ( c a + a b − b c )   :   c a ( a b + b c − c a )   :   a b ( b c + c a − a b ) {\displaystyle bc(ca+ab-bc)\ :\ ca(ab+bc-ca)\ :\ ab(bc+ca-ab)}

Construction for the equal parallelians point

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.

See also

References

  1. ^ Kimberling, Clark. "Equal Parallelians Point". Archived from the original on 16 May 2012. Retrieved 12 June 2012.
  2. ^ 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.
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