Elementary geometry stubs

"Elementary geometry stubs" typically refers to short articles or entries on topics related to elementary geometry that are found in online encyclopedias or databases, particularly Wikipedia. These stubs contain basic information about a subject but are incomplete, lacking in-depth detail or comprehensive coverage. In the context of Wikipedia, a stub is a type of article that is too short to provide substantial information on its topic, but it has the potential to be expanded by contributors.

The term "anthropomorphic polygon" isn’t widely established in mathematics or art; however, it can broadly refer to a polygon (a geometric shape with straight sides) that is designed or represented in such a way that it embodies human-like characteristics or attributes. In design, animation, and gaming, anthropomorphism is commonly used to give inanimate objects or animals human traits, emotions, or behaviors.

In geometry, an "apex" refers to the highest point or the tip of a geometric figure, particularly in the context of three-dimensional shapes. For example: 1. **Pyramids**: The apex is the top vertex of the pyramid, which is not part of the base. The sides of the pyramid rise from the base to meet at the apex.

Aristarchus's inequality is a principle related to the geometry of circles, particularly in the context of convex polygons and their tangents. The inequality asserts that for any convex polygon inscribed in a circle, the sum of the lengths of the tangents drawn from any point inside the circle to the sides of the polygon is bounded by a certain value that depends on the polygon and the radius of the circle.

An auxiliary line is a line that is added to a diagram in geometry to help in the solving of a problem or proving a theorem. It is not originally part of the figure and is typically drawn to provide additional information or to create relationships that were not previously apparent. Auxiliary lines can facilitate the construction of new angles, help to demonstrate congruence or similarity between triangles, and can make it easier to visualize geometric relationships.

Axial symmetry, also known as rotational symmetry or cylindrical symmetry, refers to a property of a shape or object where it appears the same when rotated around a particular axis. In simpler terms, if you can rotate the object about a specific line (the axis), it will look identical at various angles of rotation.

Bottema's theorem is a result in elementary geometry related to the properties of triangles and their centroids (centers of mass) associated with certain geometric transformations. Specifically, it deals with how the centroids of the segments connecting the vertices of a triangle to points on the opposite sides behave under certain conditions.

The Braikenridge–Maclaurin theorem is a result from calculus that extends the idea of Taylor series. Specifically, it provides a way to approximate a function using polynomial expressions derived from the function's derivatives at a specific point, often around zero (Maclaurin series). The theorem essentially states that if a function is sufficiently smooth (i.e., it has derivatives of all orders) at a point, then it can be expressed as an infinite series expansion in terms of that point's derivatives.

The Brocard triangle is a concept in triangle geometry related to the circumcircle and the Brocard points of a given triangle. To understand the Brocard triangle, we first need to define the Brocard points, often denoted as \( \Omega_1 \) and \( \Omega_2 \).

In geometry, a capsule is a three-dimensional shape formed by combining a cylindrical section with two hemispherical ends. Visually, it resembles a capsule or pill, which is where it gets its name. The geometric characteristics of a capsule can be defined based on parameters such as: 1. **Length**: The distance between the flat surfaces of the two hemispheres along the central axis of the cylinder.

Circle packing in a circle refers to the arrangement of smaller circles within a larger circle in such a way that the smaller circles do not overlap and are as densely packed as possible. This problem can be seen as a geometric optimization problem where the objective is to maximize the number of smaller circles that can fit within the confines of the larger circle while adhering to certain rules of arrangement. ### Key Concepts: 1. **Inner Circle**: This is the larger circle within which the smaller circles will be packed.

Circle packing in a square refers to the arrangement of circles of a specific size within a square area such that the circles do not overlap and are contained completely within the square. This is a geometrical problem that has been studied in mathematics, particularly in the fields of combinatorics and optimization. ### Key Concepts: 1. **Packing Density**: This refers to the fraction of the square's area that is occupied by the circles. The goal is often to maximize this density.

Circle packing in an equilateral triangle refers to the arrangement of circles within the confines of an equilateral triangle such that the circles touch each other and the sides of the triangle without overlapping. This geometric configuration is of interest in both mathematics and art due to its elegance and the interesting properties that arise from the arrangement.

Circle packing in an isosceles right triangle refers to the arrangement of circles (typically of equal size) within the confines of an isosceles right triangle such that the circles do not overlap and are completely contained within the triangle. In an isosceles right triangle, the two equal sides form a right angle, and the circles can be arranged in various patterns based on geometric principles and packing density.

The Crossbar Theorem is a concept in topology and combinatorial geometry. It deals with configurations of points and lines in a plane.

An eleven-point conic is a mathematical term that refers to a specific configuration involving points and projections in projective geometry, particularly in the study of conics. A conic section, or conic, is a curve obtained from the intersection of a cone with a plane. The most common types of conics are ellipses, parabolas, and hyperbolas.

The Eyeball theorem, often encountered in the context of algebraic geometry, is a humorous and informal way of illustrating certain geometric concepts involving curves and their behavior. However, it's not a standardized theorem with a formal proof in the same way as established mathematical principles. In a more specific mathematical context, the term "eyeball" might refer to visualizing properties of curves or surfaces, particularly in terms of intersections, singular points, or other geometric characteristics.

The term "GEOS circle" is often associated with geographic information systems (GIS) and refers to a circular area surrounding a specific point on the Earth's surface, typically defined by a given radius. This concept is frequently used in spatial analysis, mapping, and geolocation applications to illustrate zones of influence, proximity, or to perform geospatial queries.

Jacobi's theorem in geometry, often associated with the work of mathematician Carl Gustav Jacob Jacobi, pertains to the study of the curvature and geometric properties of surfaces. One of the key aspects of Jacobi's theorem relates to the behavior of geodesics on surfaces, particularly in the context of the stability of geodesic flow. In a more specific formulation, Jacobi's theorem can be understood in terms of the Jacobi metric on a given manifold.

In geometry, a limiting point (also known as an accumulation point or cluster point) refers to a point that can be approached by a sequence of points from a given set, such that there are points in the set arbitrarily close to it.

Moss's egg, often referred to as "Moss's green egg," is a term associated with a type of egg known for its characteristic greenish color. This is specifically observed in certain species of birds or reptiles. In ornithology, it might refer to eggs laid by some species of birds that have a mossy or greenish tint.

Pasch's theorem is a fundamental result in the field of geometry, specifically related to the properties of points and lines in a plane. It can be stated as follows: **Theorems Statement**: If a line intersects one side of a triangle and does not pass through any of the triangle's vertices, then it must intersect at least one of the other two sides of the triangle.

Plane symmetry, also known as reflectional symmetry or mirror symmetry, is a type of symmetry in which an object is invariant under reflection across a given plane. In simpler terms, if you were to "fold" an object along a plane, the two halves of the object would match perfectly. In mathematical and geometric contexts, a plane of symmetry divides an object into two mirror-image halves. For example, many organic and inorganic shapes possess at least one plane of symmetry.

A Poncelet point is a concept in projective geometry, named after the French mathematicianJean-Victor Poncelet. It refers to a specific point associated with a pair of conics (typically two ellipses or hyperbolas) that have a certain geometric relationship.

A tangential triangle, also known as a circumscribed triangle, is a type of triangle that has an incircle (a circle that is tangent to all three sides) and the center of this incircle is known as the incenter. The tangential triangle is formed when a triangle has an incircle that touches each side at exactly one point.

Tarry Point typically refers to a geographic location or area, often used to describe a point along a river or body of water where there is a notable characteristic, such as a scenic overlook, recreational area, or a point where vessels may stop or anchor. One notable example is Tarrytown, New York, which is located near the Tarry Point on the Hudson River. This area is known for its picturesque views of the river and surrounding landscape, as well as historical significance.

In anatomy, the transversal plane (also known as the transverse plane or horizontal plane) is an imaginary plane that divides the body into superior (upper) and inferior (lower) parts. This plane runs horizontally across the body, perpendicular to both the sagittal plane (which divides the body into left and right) and the coronal (frontal) plane (which divides the body into anterior (front) and posterior (back) sections).

"Woo circles" refers to the concept discussed in network marketing or multi-level marketing (MLM) contexts. It describes the idea of creating a close-knit group or community of individuals who support and promote each other's businesses, often through social media platforms. In this setting, "Woo" is typically associated with the idea of influencing or charming others, a term popularized in the context of personality strengths by the Gallup StrengthsFinder assessment.