Parry point (triangle)

In geometry, the Parry point is a special point associated with a plane triangle. It is the triangle center designated X 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.

Parry circle

Let be a plane triangle. The circle through the centroid and the two isodynamic points of is called the Parry circle of. The equation of the Parry circle in barycentric coordinates is
The center of the Parry circle is also a triangle center. It is the center designated as X in the Encyclopedia of Triangle Centers. The trilinear coordinates of the center of the Parry circle are
The center of the Parry circle is the centroid of the centers of the three circles of Apollonius, which are collinear on the Lemoine axis. The center also lies on the trilinear polar of the intersection of the Lemoine axis and the line at infinity.
The axes of the Steiner ellipse intersect the Lemoine axis on the Parry circle.

The Parry circle and the circumcircle of triangle intersect in two points. One of them is the focus of the Kiepert parabola of. The other point of intersection is called the Parry point of.
The trilinear coordinates of the Parry point are
The Parry point, the centroid and the Steiner point of are collinear.
The point of intersection of the Parry circle and the circumcircle of which is the focus of the Kiepert parabola of is also a triangle center and it is designated as X in Encyclopedia of Triangle Centers. The trilinear coordinates of this triangle center are
The focus of the Kiepert parabola, the centroid and the Tarry point of are collinear.