geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the Parry point is a special point associated with a plane

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. It is the

triangle center In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, ...

designated ''X''(111) in

Clark Kimberling Clark Kimberling (born November 7, 1942, in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer se ...

's ''

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or " centers" associated with the geometry of a triangle. This resource is hosted at the University of Evansville The University of Evansville (UE) is a priv ...

''. 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

and the two

isodynamic point In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an Inversive geometry, inversion centered at one of these points transforms the given triangle into an equilateral triang ...

s of is called the Parry circle of . The equation of the Parry circle in barycentric coordinates is

3(b^2-c^2)(c^2-a^2)(a^2-b^2)(a^2yz+b^2zx+c^2xy) + (x+y+z)\left( \sum_\text b^2c^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 In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ...

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 In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

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

of the Parry point are

\frac : \frac : \frac

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''(110) in

. The trilinear coordinates of this triangle center are

\frac : \frac : \frac

References

{{Use dmy dates, date=May 2025 Triangle centers

Parry circle

Parry point

See also

References