Equal parallelians point
Triangle center 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

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
The trilinear coordinates of the equal parallelians point of triangle △ABC are
Construction for the equal parallelians point

Reference triangle △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
References
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