geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the Parry point is a special point associated with a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. It is the

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...

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 seq ...

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. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...

. 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

Let ''ABC'' 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 surface of the figure. The same definition extends to any ...

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 inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...

s of triangle ''ABC'' is called the Parry circle of triangle ''ABC''. The equation of the Parry circle in barycentric coordinates is :

& + (x+y+z)\left( \sum_\text b^2c^2(b^2-c^2)(b^2+c^2-2a^2)x\right) =0 \end

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(b^2-c^2)(b^2+c^2-2a^2)

Parry point

The Parry circle and the

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

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 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 Parry point are :

a/(2a^2-b^2-c^2) : b/(2b^2-c^2-a^2) : c/(2c^2-a^2-b^2)

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

. The trilinear coordinates of this triangle center are :

a/(b^2-c^2) : b/(c^2-a^2) : c/(a^2-b^2)

References

{{reflist Triangle centers

Parry circle

Parry point

See also

References