Equal parallelians point

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
[[File:EqualParalleliansPoint.svg|thumb|250px|

]] 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 $$bc(ca+ab-bc) \ : \ ca(ab+bc-ca) \ : \ ab(bc+ca-ab)$$

Construction for the equal parallelians point
[[File:ConstructionOfEqualParalleliansPoint.svg|thumb|250px|Construction of the equal parallelians point.

]] Let $△ABC$ 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 $A'A", B'B", C'C"$ respectively. Then the lines $A, B, C$ concur at the equal parallelians point of $△ABC$.