Equal parallelians point
In geometry, the equal parallelians point is a special point associated with a plane triangle. It is a triangle center and it is denoted by X 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
The equal parallelians point of triangle is a point in the plane of such that the three line segments through parallel to the sidelines of and having endpoints on these sidelines have equal lengths.Trilinear coordinates
The trilinear coordinates of the equal parallelians point of triangle areConstruction for the equal parallelians point
Let be the anticomplementary triangle of triangle. Let the internal bisectors of the angles at the vertices of meet the opposite sidelines at respectively. Then the lines concur at the equal parallelians point of.As a result, the equal parallelians point is the isotomic conjugate of the incenter of and with respect to the anticomplementary triangle.