# Thomson cubic

In geometry, the **Thomson cubic** of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent.

## See also[edit]

## References[edit]

Weisstein, Eric W. "Thomson cubic". *MathWorld*.

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In geometry, the **Thomson cubic** of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent.

Weisstein, Eric W. "Thomson cubic". *MathWorld*.

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1. Geometry – Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

2. Triangle – A triangle is a polygon with three edges and three vertices. It is one of the shapes in geometry. A triangle with vertices A, B, and C is denoted △ A B C, in Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane. This article is about triangles in Euclidean geometry except where otherwise noted, triangles can be classified according to the lengths of their sides, An equilateral triangle has all sides the same length. An equilateral triangle is also a polygon with all angles measuring 60°. An isosceles triangle has two sides of equal length, some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides. The latter definition would make all equilateral triangles isosceles triangles, the 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles. A scalene triangle has all its sides of different lengths, equivalently, it has all angles of different measure. Hatch marks, also called tick marks, are used in diagrams of triangles, a side can be marked with a pattern of ticks, short line segments in the form of tally marks, two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is no more than 3 ticks. Similarly, patterns of 1,2, or 3 concentric arcs inside the angles are used to indicate equal angles, triangles can also be classified according to their internal angles, measured here in degrees. A right triangle has one of its interior angles measuring 90°, the side opposite to the right angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs or catheti of the triangle, special right triangles are right triangles with additional properties that make calculations involving them easier. One of the two most famous is the 3–4–5 right triangle, where 32 +42 =52, in this situation,3,4, and 5 are a Pythagorean triple. The other one is a triangle that has 2 angles that each measure 45 degrees. Triangles that do not have an angle measuring 90° are called oblique triangles, a triangle with all interior angles measuring less than 90° is an acute triangle or acute-angled triangle. If c is the length of the longest side, then a2 + b2 > c2, a triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. If c is the length of the longest side, then a2 + b2 < c2, a triangle with an interior angle of 180° is degenerate

3. Locus (mathematics) – In geometry, a locus is a set of points, whose location satisfies or is determined by one or more specified conditions. Until the beginning of 20th century, a shape was not considered as an infinite set of points, rather. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a distance of a fixed point. In contrast to the view, the old formulation avoids considering infinite collections. Once set theory became the universal basis over which the mathematics is built. Examples from plane geometry include, The set of points equidistant from two points is a perpendicular bisector to the segment connecting the two points. The set of points equidistant from two lines cross is the angle bisector. All conic sections are loci, Parabola, the set of points equidistant from a single point, Circle, the set of points for which the distance from a single point is constant. The set of points for each of which the ratio of the distances to two given foci is a constant is referred to as a Circle of Apollonius. Hyperbola, the set of points for each of which the value of the difference between the distances to two given foci is a constant. Ellipse, the set of points for each of which the sum of the distances to two given foci is a constant, the circle is the special case in which the two foci coincide with each other. Other examples of loci appear in areas of mathematics. For example, in dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps. Proof that all the points on the given shape satisfy the conditions and we find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k=3, A and B as the fixed points and it is the circle of Apollonius defined by these values of k, A, and B. A triangle ABC has a side with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal and we choose an orthonormal coordinate system such that A, B. C is the third vertex

4. Vertex (geometry) – In geometry, a vertex is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. A vertex is a point of a polygon, polyhedron, or other higher-dimensional polytope. However, in theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. However, a smooth approximation to a polygon will also have additional vertices. A polygon vertex xi of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at x and x, there are two types of principal vertices, ears and mouths. A principal vertex xi of a simple polygon P is called an ear if the diagonal that bridges xi lies entirely in P, according to the two ears theorem, every simple polygon has at least two ears. A principal vertex xi of a simple polygon P is called a mouth if the diagonal lies outside the boundary of P. Any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges and this equation is known as Eulers polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces, for example, a cube has 12 edges and 6 faces, and hence 8 vertices

5. Concurrent lines – In geometry, three or more lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. The point where the three altitudes meet is the orthocenter, angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter, medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid, perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter, other sets of lines associated with a triangle are concurrent as well. For example, Any median is concurrent with two other area bisectors each of which is parallel to a side, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle, a splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle, Any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter, and each triangle has one, two, or three of these lines. Thus if there are three of them, they concur at the incenter, the Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangles first Brocard triangle. The Napoleon points and generalizations of them are points of concurrency, a generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line, in the case in which the original triangle has no angle greater than 120°, this point is also the Fermat point. The two bimedians of a quadrilateral and the segment joining the midpoints of the diagonals are concurrent and are all bisected by their point of intersection. In a tangential quadrilateral, the four angle bisectors concur at the center of the incircle, other concurrencies of a tangential quadrilateral are given here. In a cyclic quadrilateral, four segments, each perpendicular to one side. These line segments are called the maltitudes, which is an abbreviation for midpoint altitude and their common point is called the anticenter. If the successive sides of a hexagon are a, b, c, d, e, f. If a hexagon has a conic, then by Brianchons theorem its principal diagonals are concurrent. Concurrent lines arise in the dual of Pappuss hexagon theorem, for each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side

6. Cubic plane curve – Here F is a non-zero linear combination of the third-degree monomials x3, y3, z3, x2y, x2z, y2x, y2z, z2x, z2y, xyz. These are ten in number, therefore the cubic curves form a space of dimension 9. Each point P imposes a single linear condition on F, if we ask that C pass through P, if two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties, see Cayley–Bacharach theorem. A cubic curve may have a point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection and this can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C, the intersections are then counted by Bézouts theorem. However, only three of these points may be real, so that the others cannot be seen in the projective plane by drawing the curve. The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points, the real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two ovals, the other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, a non-singular cubic defines an elliptic curve, over any field K for which it has a point defined. This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form, there are many cubic curves that have no such point, for example when K is the rational number field. The singular points of a plane cubic curve are quite limited, one double point. A reducible plane cubic curve is either a conic and a line or three lines, and accordingly have two points or a tacnode, or up to three double points or a single triple point if three lines. Suppose that ABC is a triangle with sidelengths a = | BC |, b = | CA |, relative to ABC, many named cubics pass through well-known points. Examples shown below use two kinds of coordinates, trilinear and barycentric. To convert from trilinear to barycentric in an equation, substitute as follows, x ↦ bcx, y ↦ cay, z ↦ abz, to convert from barycentric to trilinear, use x ↦ ax, y ↦ by. Many equations for cubics have the form f + f + f =0, in the examples below, such equations are written more succinctly in cyclic sum notation, like this, [cyclic sum f =0. The cubics listed below can be defined in terms of the conjugate, denoted by X*. Let LA be the reflection of line XA about the angle bisector of angle A