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Equal Parallelians Point

Through a point in the plane of a triangle , draw parallelians through a point as illustrated above. Then there exist four points for which , i.e., for which the segments of the parallels have equal length.

To restrict these four points, let the length of be considered negative if lies on the extension of beyond and lies on the extension of beyond , and positive otherwise. Define the lengths of the other two parallelians to be signed in the analogous manner. With this sign convention, there is a unique point for which the signed parallelians have equal length. This point is called the equal parallelians point of .

It has equivalent triangle center functions

 (1) (2)

and is Kimberling center (Kimberling 1998, p. 104).

The length of the equal parallelians is

 (3) (4)

As is true for general parallelians, those for the lie on an ellipse. The center of this ellipse has triangle center function

 (5)

which is not a Kimberling center.

The equal parallelians point is also the perspector of the incentral triangle and anticomplementary triangle of (Kimberling 1998, p. 105).

Congruent Isoscelizers Point, Parallelian

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References

Bier, S. "Equilateral Triangles Intercepted by Oriented Parallelians." Forum Geom. 1, 25-32, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200105index.html.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Equal Parallelians Point." http://faculty.evansville.edu/ck6/tcenters/recent/eqparal.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(192)=X(1)-Ceva Conjugate of X(2) (Equal Parallelians Point)." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X192.Yiu, P. "Geometric Constructions VII: Solution of GC15." Math. and Informatics Quart. http://www.math.fau.edu/yiu/MIQGeomConstructions7.ps.

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Equal Parallelians Point

Cite this as:

Weisstein, Eric W. "Equal Parallelians Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EqualParalleliansPoint.html