Equal parallelians point
{{short description|Triangle center}}
In geometry, the equal parallelians point{{cite web|last=Kimberling |first=Clark |title=Equal Parallelians Point |url=http://faculty.evansville.edu/ck6/tcenters/recent/eqparal.html |accessdate=12 June 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120516140424/http://faculty.evansville.edu/ck6/tcenters/recent/eqparal.html |archivedate=16 May 2012 }}{{cite web|last=Weisstein|first=Eric|title=Equal Parallelians Point|url=http://mathworld.wolfram.com/EqualParalleliansPoint.html|work=MathWorld--A Wolfram Web Resource|accessdate=12 June 2012}} (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.{{cite web |last=Kimberling |first=Clark |title=Encyclopedia of Triangle Centers |url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |accessdate=12 June 2012 |url-status=dead |archiveurl=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |archivedate=19 April 2012 }} There is a reference to this point in one of Peter Yff's notebooks, written in 1961.
Definition
[[File:EqualParalleliansPoint.svg|thumb|250px|
{{legend-line|solid black|Reference triangle {{math|△ABC}}}}
{{legend-line|solid red|Line segments of equal length, parallel to the sidelines of {{math|△ABC}}}}
]]
The equal parallelians point of triangle {{math|△ABC}} is a point {{mvar|P}} in the plane of {{math|△ABC}} such that the three line segments through {{mvar|P}} parallel to the sidelines of {{math|△ABC}} and having endpoints on these sidelines have equal lengths.
Trilinear coordinates
The trilinear coordinates of the equal parallelians point of triangle {{math|△ABC}} are
Construction for the equal parallelians point
[[File:ConstructionOfEqualParalleliansPoint.svg|thumb|250px|Construction of the equal parallelians point.
{{legend-line|solid black|Reference triangle {{math|△ABC}}}}
{{legend-line|dashed green 2px|Internal bisectors of {{math|△ABC}} (intersect opposite sides at {{mvar|A", B", C"}})}}
{{legend-line|solid cyan|Anticomplementary triangle {{math|△A'B'C' }} of {{math|△ABC}}}}
{{legend-line|dashed blue 2px|Lines ({{mvar|A'A", B'B", C'C"}}) concurrent at the equal parallelians point}}
]]
Let {{math|△A'B'C' }} be the anticomplementary triangle of triangle {{math|△ABC}}. Let the internal bisectors of the angles at the vertices {{mvar|A, B, C}} of {{math|△ABC}} meet the opposite sidelines at {{mvar|A", B", C"}} respectively. Then the lines {{mvar|A'A", B'B", C'C"}} concur at the equal parallelians point of {{math|△ABC}}.
See also
References
{{Reflist}}