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. [1]
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 [2]
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
where
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. [3] The other point of intersection is called the Parry point of triangle ABC.
The trilinear coordinates of the Parry point are
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
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.
In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) in Clark Kimberling's Encyclopedia of Triangle Centers, and is named after Karl Wilhelm Feuerbach.
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
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.
In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.
In geometry, the trilinear coordinatesx:y:z 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 x:y is the ratio of the perpendicular distances from the point to the sides opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A.
In geometry, a centre of an object is a point in some sense in the middle of the object. According to the specific definition of center taken into consideration, an object might have no center. If geometry is regarded as the study of isometry groups then a center is a fixed point of all the isometries which move the object onto itself.
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 distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid; every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg (1885).
In triangle geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.
In geometry, a triangle center 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 example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal Journal de mathématiques pures et appliquées.
In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.
In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle ABC is the center of gravity of a homogeneous wire frame in the shape of triangle ABC. The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.
In triangle geometry, the Steiner point is a particular point associated with a triangle. It is a triangle center and it is designated as the center X(99) in Clark Kimberling's Encyclopedia of Triangle Centers. Jakob Steiner (1796–1863), Swiss mathematician, described this point in 1826. The point was given Steiner's name by Joseph Neuberg in 1886.
In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a computers-in-mathematics workshop at Phillips Exeter Academy in 1986. This is one of the recent triangle centers, having been discovered only in 1986, unlike the classical triangle centers like centroid, incenter, and Steiner point.
In geometry, central lines are certain special straight lines that lie in the plane of a triangle. The special property that distinguishes a straight line as a central line is manifested via the equation of the line in trilinear coordinates. This special property is related to the concept of triangle center also. The concept of a central line was introduced by Clark Kimberling in a paper published in 1994.
The Clawson point is a special point in a planar triangle defined by the trilinear coordinates , where are the interior angles at the triangle vertices . It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly.
