Equal parallelians point

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]

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]

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

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]

• Congruent isoscelizers point