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 the Parry circle are named in honor of the English geometer Cyril Parry, who studied them in the early 1990s.
Definition
The Parry point is defined as one of the two intersection points of the Parry circle and the circumcircle of a given triangle. The other intersection point is the focus of the Kiepert parabola, which is designated as triangle center X(110).
Parry Circle
The Parry circle is the circle that passes through the centroid and the two isodynamic points of a triangle. It also passes through several other notable triangle centers. The center of the Parry circle is itself a triangle center, designated X(351).
Coordinates
The trilinear coordinates of the Parry point are:
a/(2a² − b² − c²) : b/(2b² − c² − a²) : c/(2c² − a² − b²)
Properties
- The Parry point, the centroid, and the Steiner point of a triangle are collinear.
- The Parry point lies on the circumcircle of the triangle, distinguishing it from interior triangle centers.
- The axes of the Steiner ellipse intersect the Lemoine axis on the Parry circle.
