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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.
Contents
Definition
The equal parallelians point of triangle ABC is a point P in the plane of triangle ABC such that the three 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
( bc ( ca + ab – bc ) : ca ( ab + bc – ca ) : ab ( bc + ca – ab ) )Construction for 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 triangle ABC meet the opposite sidelines at A'', B'', C'' respectively. Then the lines A'A'', B'B'' and C'C'' concur at the equal parallelians point of triangle ABC.