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

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The equal parallelians point of triangle Template:Math is a point Template:Mvar in the plane of Template:Math such that the three line segments through Template:Mvar parallel to the sidelines of Template:Math and having endpoints on these sidelines have equal lengths.

Trilinear coordinates

The trilinear coordinates of the equal parallelians point of triangle Template:Math are $$bc(ca+ab-bc):ca(ab+bc-ca):ab(bc+ca-ab)$$

Construction for the equal parallelians point

Construction of the equal parallelians point. Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

Let Template:Math be the anticomplementary triangle of triangle Template:Math. Let the internal bisectors of the angles at the vertices Template:Mvar of Template:Math meet the opposite sidelines at Template:Mvar respectively. Then the lines Template:Mvar concur at the equal parallelians point of Template:Math.

See also

• Congruent isoscelizers point