Parry point (triangle)

  Reference triangle △ABC

  Circumcircle of △ABC

  Apollonian circles (intersect at the isodynamic points J, K)

  Parry circle (through J, K and centroid G)

The Parry circle intersects the circumcircle at two points: the focus of the Kiepert parabola, and the Parry point.

Let △ABC be a plane triangle. The circle through the centroid and the two isodynamic points of △ABC is called the Parry circle of △ABC. The equation of the Parry circle in barycentric coordinates is[2]

$${\displaystyle 3(b^{2}-c^{2})(c^{2}-a^{2})(a^{2}-b^{2})(a^{2}yz+b^{2}zx+c^{2}xy)+(x+y+z)\left(\sum _{\text{cyclic}}b^{2}c^{2}(b^{2}-c^{2})(b^{2}+c^{2}-2a^{2})x\right)=0}$$

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

$${\displaystyle a(b^{2}-c^{2})(b^{2}+c^{2}-2a^{2}):b(c^{2}-a^{2})(c^{2}+a^{2}-2b^{2}):c(a^{2}-b^{2})(a^{2}+b^{2}-2c^{2})}$$

The Parry circle and the circumcircle of triangle △ABC intersect in two points. One of them is a focus of the Kiepert parabola of △ABC.[3] The other point of intersection is called the Parry point of △ABC.

The trilinear coordinates of the Parry point are

$${\displaystyle {\frac {a}{2a^{2}-b^{2}-c^{2}}}:{\frac {b}{2b^{2}-c^{2}-a^{2}}}:{\frac {c}{2c^{2}-a^{2}-b^{2}}}}$$

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

$${\displaystyle {\frac {a}{b^{2}-c^{2}}}:{\frac {b}{c^{2}-a^{2}}}:{\frac {c}{a^{2}-b^{2}}}}$$

• Lester circle