Circle (3)

The circle can be defined as the shape consisting of all points in a plane that are a fixed distance, called the radius, away from a fixed point, called the center. It can also be defined as the set of points in a plane that have a constant ratio of distances to two fixed foci. These definitions are connected by the fact that the distance from any point on the circle to its center is equal to its radius. Apollonius of Perga showed that a circle can be drawn about two points by using the angle bisector theorem to prove that a line segment connecting the two points and passing through a third point on the circle will always bisect the interior and exterior angles formed by the two points and the third point. The set of all points satisfying this condition forms a circle, with the line segment serving as a diameter.

The circle is a geometric shape that has captured the fascination of humans for centuries. It is defined as a collection of points in a plane that are all the same distance from a central point. In mathematics, circles can be described using various equations. One common equation is the equation for a circle with a center at point c and radius r, which is written as |z-c|=r. Another equation for a circle is the slightly generalized equation pz|z|+gz+|gz|=q, where p, q are real numbers and g is a complex number. This equation is known as a generalized circle and can represent either a true circle or a line.

One interesting property of circles is that they have a high degree of symmetry. Every line that passes through the center of a circle forms a line of reflection symmetry, and the circle has rotational symmetry around the center for every angle. In addition, all circles are similar to one another. The relationship between a circle's circumference and radius is also notable - they are proportional to each other. The area enclosed within a circle is proportional to the square of its radius. These constants of proportionality are 2π and π, respectively.

There are many theorems and properties that pertain to circles. For example, the tangent line through a point on a circle is perpendicular to the diameter passing through that point. The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle, and an inscribed angle that is subtended by a diameter is a right angle. The diameter of a circle is the longest chord, and the sum of the squared lengths of any two chords intersecting at right angles at a given point is a constant value. There are also several ways to construct a circle using a compass and straightedge. One method is to use the center of the circle and a point on the circle. Another method involves constructing the midpoint of a diameter and drawing the circle with that point as the center. Finally, the circle of Apollonius can be constructed by finding the points that satisfy a certain ratio of distances to two fixed points. This circle may either be a true circle or a line, depending on the positions of the fixed points.

A circle is a geometric shape with a round and circular edge. It is defined by a set of points in a plane that are all an equal distance from a central point. The distance from the center to any point on the circle is known as the radius. A line drawn through the center of the circle and connecting two points on the circle is called the diameter. A circle is highly symmetrical, with every line through its center forming a line of reflection symmetry. It also has rotational symmetry around its center for every angle.

A circle can be constructed by drawing a radius from its center to a point on the circle, or by drawing the midpoint of a diameter and then constructing a circle that passes through one of the endpoints of the diameter. A circle can also be defined by two fixed points known as foci, and the set of points that have a constant ratio of distances to these two foci. This circle is known as the circle of Apollonius.

A special kind of triangle, called a tangential triangle, can have a circle inscribed within it that is tangent to each of its sides. Similarly, a cyclic triangle is a triangle that can have a circle circumscribed around it that passes through each of its vertices. A regular polygon or any triangle is both a tangential and cyclic polygon. A curve that is inscribed in a circle by tracing a fixed point on a smaller circle that rolls within and is tangent to the larger circle is called a hypocycloid.

The circle is a shape with the largest area for a given length of perimeter, as stated by the Isoperimetric inequality. It is a highly symmetrical shape, with every line through its center forming a line of reflection symmetry and having rotational symmetry around its center for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is known as the circle group T. All circles are similar, and the circumference and radius of a circle are proportional, with constants of proportionality being 2π and π, respectively. The area enclosed and the square of the radius are also proportional. The circle that is centered at the origin with radius 1 is known as the unit circle. When considered as a great circle of the unit sphere, it becomes the Riemannian circle. For any three points that are not all on the same line, there exists a unique circle that passes through them. In Cartesian coordinates, explicit formulae for the coordinates of the center of the circle and the radius in terms of the coordinates of the three given points can be derived. This is known as a circumcircle.

A circle can be defined as the set of all points in a plane that are a fixed distance from a given point, known as the center. This distance is known as the radius of the circle. There are many properties and constructions that involve circles. For example, the incircle of a triangle is a circle that is tangent to all three sides of the triangle. The circumcircle of a triangle is a circle that passes through all three vertices of the triangle. A tangential polygon is a convex polygon that has a circle inscribed in it that is tangent to all sides of the polygon. Cyclic polygons are convex polygons that have a circle circumscribed around them that passes through all vertices of the polygon. A hypocycloid is a curve that is inscribed in a circle by tracing a fixed point on a smaller circle that rolls within and is tangent to the larger circle. In addition, circles can be defined in different ways depending on the distance measure used. For example, in p-norm, the distance between two points is determined by a weighted sum of the distances between those points. In Euclidean geometry, the p-norm is defined as p=2, which results in the familiar distance formula. In taxicab geometry, p=1, resulting in a distance measure known as the taxicab distance. This results in circles being defined as squares with sides oriented at a 45 degree angle to the coordinate axes. The Chebyshev distance, also known as the L∞ metric, can also be used to define circles as squares with sides parallel to the coordinate axes.

The concept of the circle has been integral to the art and culture of various civilizations throughout history. The circle represents many spiritual and sacred concepts, such as unity, infinity, and balance, and has been conveyed through symbols like the compass and halo. However, the problem of "squaring the circle," or constructing a square with the same area as a given circle using only a compass and straightedge, has been proven to be impossible due to the transcendental nature of pi. Despite this, the topic continues to fascinate some enthusiasts of pseudomath.