Parry point (triangle)

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 honor of the English geometer Cyril Parry, who studied them in the early 1990s.

Parry circle

Template:Legend-line Template:Legend-line Template:Legend striped Template:Legend-line The Parry circle intersects the circumcircle at two points: the focus of the Kiepert parabola, and the Parry point.

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

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

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 Template:Math intersect in two points. One of them is a focus of the Kiepert parabola of Template:Math. The other point of intersection is called the Parry point of Template:Math.

The trilinear coordinates of the Parry point are

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

The point of intersection of the Parry circle and the circumcircle of Template:Math which is a focus of the Kiepert hyperbola of Template:Math 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

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

See also

• Lester circle