Equal parallelians point explained

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]

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.^[1]

Trilinear coordinates

The trilinear coordinates of the equal parallelians point of triangle are$$bc(ca+ab-bc)\; \backslash \; :\; \backslash \; ca(ab+bc-ca)\; \backslash \; :\; \backslash \; ab(bc+ca-ab)$$

Construction 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 .^[2]

See also

• Congruent isoscelizers point

Notes and References

1. Web site: Kimberling . Clark . Equal Parallelians Point . 12 June 2012 . dead . https://web.archive.org/web/20120516140424/http://faculty.evansville.edu/ck6/tcenters/recent/eqparal.html . 16 May 2012 .
2. Web site: Weisstein. Eric. Equal Parallelians Point. MathWorld--A Wolfram Web Resource. 12 June 2012.
3. Web site: Kimberling . Clark . Encyclopedia of Triangle Centers . 12 June 2012 . dead . https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html . 19 April 2012 .