Yff center of congruence#Yff central triangle

An isoscelizer of an angle {{mvar|A}} in a triangle {{math|△ABC}} is a line through points {{math|P{{sub|1}}, Q{{sub|1}}}}, where {{math|P{{sub|1}}}} lies on {{mvar|AB}} and {{math|Q{{sub|1}}}} on {{mvar|AC}}, such that the triangle {{math|△AP{{sub|1}}Q{{sub|1}}}} is an isosceles triangle. An isoscelizer of angle {{mvar|A}} is a line perpendicular to the bisector of angle {{mvar|A}}. Isoscelizers were invented by Peter Yff in 1963.{{cite web|last=Weisstein|first=Eric W.|title=Isoscelizer|url=http://mathworld.wolfram.com/Isoscelizer.html|publisher=MathWorld--A Wolfram Web Resource.|access-date=30 May 2012}}

[[File:Yff center of congruence.svg|thumb|

{{legend-line|solid black|Reference triangle {{math|△ABC}}}}

{{legend inline|#2fcc27|{{math|△A'P{{sub|2}}Q{{sub|3}} ≅ }}}}

{{legend inline|#4f46f2|{{math|△Q{{sub|1}}B'P{{sub|3}} ≅ }}}}

{{legend inline|#ff9a36|{{math|△P{{sub|1}}Q{{sub|2}}C' ≅}}}}

{{legend|#ff5252|{{math|△A'B'C' }} (Yff central triangle)}}

]]

Let {{math|△ABC}} be any triangle. Let {{math|P{{sub|1}}Q{{sub|1}}}} be an isoscelizer of angle {{mvar|A}}, {{math|P{{sub|2}}Q{{sub|2}}}} be an isoscelizer of angle {{mvar|B}}, and {{math|P{{sub|3}}Q{{sub|3}}}} be an isoscelizer of angle {{mvar|C}}. Let {{math|△A'B'C' }} be the triangle formed by the three isoscelizers. The four triangles {{math|△A'P{{sub|2}}Q{{sub|3}}, △Q{{sub|1}}B'P{{sub|3}}, △P{{sub|1}}Q{{sub|2}}C',}} and {{math|△A'B'C' }} are always similar.

There is a unique set of three isoscelizers {{math|P{{sub|1}}Q{{sub|1}}, P{{sub|2}}Q{{sub|2}}, P{{sub|3}}Q{{sub|3}}}} such that the four triangles {{math|△A'P{{sub|2}}Q{{sub|3}}, △Q{{sub|1}}B'P{{sub|3}}, △P{{sub|1}}Q{{sub|2}}C',}} and {{math|△A'B'C' }} are congruent. In this special case {{math|△A'B'C' }} formed by the three isoscelizers is called the Yff central triangle of {{math|△ABC}}.{{cite web|last=Weisstein|first=Eric W|title=Yff central triangle|url=http://mathworld.wolfram.com/YffCentralTriangle.html|publisher=MathWorld--A Wolfram Web Resource.|access-date=30 May 2012}}

The circumcircle of the Yff central triangle is called the Yff central circle of the triangle.

File:Yff center of congruence.gif

Let {{math|△ABC}} be any triangle. Let {{math|P{{sub|1}}Q{{sub|1}}, P{{sub|2}}Q{{sub|2}}, P{{sub|3}}Q{{sub|3}}}} be the isoscelizers of the angles {{mvar|A, B, C}} such that the triangle {{math|△A'B'C' }} formed by them is the Yff central triangle of {{math|△ABC}}. The three isoscelizers {{math|P{{sub|1}}Q{{sub|1}}, P{{sub|2}}Q{{sub|2}}, P{{sub|3}}Q{{sub|3}}}} are continuously parallel-shifted such that the three triangles {{math|△A'P{{sub|2}}Q{{sub|3}}, △Q{{sub|1}}B'P{{sub|3}}, △P{{sub|1}}Q{{sub|2}}C' }} are always congruent to each other until {{math|△A'B'C' }} formed by the intersections of the isoscelizers reduces to a point. The point to which {{math|△A'B'C' }} reduces to is called the Yff center of congruence of {{math|△ABC}}.

File:Yff central triangle and its excircles.svg of the Yff central triangle of {{math|△ABC}}.]]

• The trilinear coordinates of the Yff center of congruence are $$$$

\sec\frac{A}{2} : \sec\frac{B}{2} : \sec\frac{C}{2}

• Any triangle {{math|△ABC}} is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of {{math|△ABC}}.
• Let {{mvar|I}} be the incenter of {{math|△ABC}}. Let {{mvar|D}} be the point on side {{mvar|BC}} such that {{math|1=∠BID = ∠DIC}}, {{mvar|E}} a point on side {{mvar|CA}} such that {{math|1=∠CIE = ∠EIA}}, and {{mvar|F}} a point on side {{mvar|AB}} such that {{math|1=∠AIF = ∠FIB}}. Then the lines {{mvar|AD, BE, CF}} are concurrent at the Yff center of congruence. This fact gives a geometrical construction for locating the Yff center of congruence.{{cite web|last=Kimberling|first=Clark|title=X(174) = Yff Center of Congruence|url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html|access-date=2 June 2012}}
• A computer assisted search of the properties of the Yff central triangle has generated several interesting results relating to properties of the Yff central triangle.{{cite journal|last=Dekov|first=Deko|title=Yff Center of Congruence|journal=Journal of Computer-Generated Euclidean Geometry|year=2007|volume=37|pages=1–5|url=http://www.docstoc.com/docs/70786195/Yff-Center-of-Conguence|access-date=30 May 2012}}

File:Yff Centre Generalisation.png

The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point {{mvar|P}} in the plane of a triangle {{math|△ABC}}. Then points {{mvar|D, E, F}} are taken on the sides {{mvar|BC, CA, AB}} such that

$$\backslash angle\; BPD\; =\; \backslash angle\; DPC,\; \backslash quad\; \backslash angle\; CPE\; =\; \backslash angle\; EPA,\; \backslash quad\; \backslash angle\; APF\; =\; \backslash angle\; FPB.$$

The generalization asserts that the lines {{mvar|AD, BE, CF}} are concurrent.

• Congruent isoscelizers point
• Central triangle

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Category:Triangle centers