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A topological space M satisfying some separability (i.e., it is a T2-space) and countability (i.e., it is a paracompact space) conditions such that every point p in M has a ...

There are four completely different definitions of the so-called Apollonius circles: 1. The set of all points whose distances from two fixed points are in a constant ratio ...

Given three objects, each of which may be a point, line, or circle, draw a circle that is tangent to each. There are a total of ten cases. The two easiest involve three ...

Draw the perpendicular line from the intersection of the two small semicircles in the arbelos. The two circles C_1 and C_2 tangent to this line, the large semicircle, and ...

The cosine function cosx is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let theta be an ...

Generally, a face is a component polygon, polyhedron, or polytope. A two-dimensional face thus has vertices and edges, and can be used to make cells. More formally, a face is ...

The keratoid cusp is quintic algebraic curve defined by y^2=x^2y+x^5. (1) It has a ramphoid cusp at the origin, horizontal tangents at (0,0) and (-6/(25),(108)/(3125)), and a ...

The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular ...

A polygon which has both a circumcircle (which touches each vertex) and an incircle (which is tangent to each side). All triangles are bicentric with R^2-x^2=2Rr, (1) where R ...

A bitangent is a line that is tangent to a curve at two distinct points. Aa general plane quartic curve has 28 bitangents in the complex projective plane. However, as shown ...