Parry point (triangle)
{{short description|Triangle center}}
In geometry, the Parry point is a special point associated with a plane triangle. It is the triangle center designated X(111) in Clark Kimberling's Encyclopedia of Triangle Centers. The Parry point and Parry circle are named in honour of the English geometer Cyril Parry, who studied them in the early 1990s.{{cite web|last=Kimberling|first=Clark|title=Parry point|url=http://faculty.evansville.edu/ck6/tcenters/recent/parry.html|accessdate=29 May 2012}}
Parry circle
[[File:Parry point.svg|thumb|275px|
{{legend-line|solid black|Reference triangle {{math|△ABC}}}}
{{legend-line|solid orange|Circumcircle of {{math|△ABC}}}}
{{Legend striped|#e77aff|#6ebeff|Apollonian circles (intersect at the isodynamic points {{mvar|J, K}})|outline=silver}}
{{legend-line|solid red|Parry circle (through {{mvar|J, K}} and centroid {{mvar|G}})}}
The Parry circle intersects the circumcircle at two points: the focus of the Kiepert parabola, and the Parry point.]]
Let {{math|△ABC}} be a plane triangle. The circle through the centroid and the two isodynamic points of {{math|△ABC}} is called the Parry circle of {{math|△ABC}}. The equation of the Parry circle in barycentric coordinates is{{cite journal|last=Yiu|first=Paul|title=The Circles of Lester, Evans, Parry, and Their Generalizations|journal=Forum Geometricorum|year=2010|volume=10|pages=175–209|url=http://forumgeom.fau.edu/FG2010volume10/FG201020.pdf|accessdate=29 May 2012|archive-date=7 October 2021|archive-url=https://web.archive.org/web/20211007130241/https://forumgeom.fau.edu/FG2010volume10/FG201020.pdf|url-status=dead}}
3(b^2-c^2)(c^2-a^2)(a^2-b^2)(a^2yz+b^2zx+c^2xy) + (x+y+z)\left( \sum_\text{cyclic} b^2c^2(b^2-c^2)(b^2+c^2-2a^2)x\right) =0
The center of the Parry circle is also a triangle center. It is the center designated as X(351) in the Encyclopedia of Triangle Centers. The trilinear coordinates of the center of the Parry circle are
a(b^2-c^2)(b^2+c^2-2a^2) : b(c^2-a^2)(c^2+a^2-2b^2) : c(a^2-b^2)(a^2+b^2-2c^2)
Parry point
The Parry circle and the circumcircle of triangle {{math|△ABC}} intersect in two points. One of them is the focus of the Kiepert parabola of {{math|△ABC}}.{{cite web|last=Weisstein|first=Eric W|title=Parry Point|url=http://mathworld.wolfram.com/ParryPoint.html|publisher=MathWorld—A Wolfram Web Resource.|accessdate=29 May 2012}} The other point of intersection is called the Parry point of {{math|△ABC}}.
The trilinear coordinates of the Parry point are
The point of intersection of the Parry circle and the circumcircle of {{math|△ABC}} which is the focus of the Kiepert parabola of {{math|△ABC}} is also a triangle center and it is designated as X(110) in Encyclopedia of Triangle Centers. The trilinear coordinates of this triangle center are