Parry point (triangle)

Parry circle

Let ABC be a plane triangle. The circle through the centroid and the two isodynamic points of triangle ABC is called the Parry circle of triangle ABC. The equation of the Parry circle in barycentric coordinates is

3 ( b 2 − c 2 ) ( c 2 − a 2 ) ( a 2 − b 2 ) ( a 2 y z + b 2 z x + c 2 x y ) + ( x + y + z ) ( ∑ cyclic b 2 c 2 ( b 2 − c 2 ) ( b 2 + c 2 − 2 a 2 ) x ) = 0

The center of the Parry circle is also a triangle center. It is the center designated as X(351) in Encyclopedia of Triangle Centers. The trilinear coordinates of the center of the Parry circle are

f( a, b, c ) : f ( b , c, a ) : f ( c, a, b ), where f ( a , b, c ) = a ( b² − c² ) ( b² + c² − 2a² )

Parry point

The Parry circle and the circumcircle of triangle ABC intersect in two points. One of them is a focus of the Kiepert parabola of triangle ABC. The other point of intersection is called the Parry point of triangle ABC.

The trilinear coordinates of the Parry point are

( a / ( 2 a² − b² − c² ) : b / ( 2 b² − c² − a² ) : c / ( 2 c² − a² − b² ) )

The point of intersection of the Parry circle and the circumcircle of triangle ABC which is a focus of the Kiepert hyperbola of triangle 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

( a / ( b² − c² ) : b / ( b² − a² ) : c / ( a² − b² ) )