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]
Definition[]
The equal parallelians point of triangle ABC is a point P in the plane of triangle ABC such that the three segments through P parallel to the sidelines of ABC and having endpoints on these sidelines have equal lengths.[1]
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[]
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 triangle ABC meet the opposite sidelines at A'', B'', C'' respectively. Then the lines A'A'', B'B'' and C'C'' concur at the equal parallelians point of triangle ABC.[2]
See also[]
References[]
- ^ a b c Kimberling, Clark. "Equal Parallelians Point". Archived from the original on 16 May 2012. Retrieved 12 June 2012.
- ^ a b Weisstein, Eric. "Equal Parallelians Point". MathWorld--A Wolfram Web Resource. Retrieved 12 June 2012.
- ^ Kimberling, Clark. "Encyclopedia of Triangle Centers". Archived from the original on 19 April 2012. Retrieved 12 June 2012.
- Triangle centers