Problem of Apollonius

Apollonius of Perga posed and solved this famous problem in his work. Three given circles generically have eight different circles that are tangent to them. The solution to this problem is known as the Apollonian gasket, after the Greek mathematician who solved it in the 4th century AD.

About Problem of Apollonius in brief

Apollonius of Perga posed and solved this famous problem in his work. Three given circles generically have eight different circles that are tangent to them. In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas. François Viète’s approach, which uses simpler limiting cases to solve more complicated ones, is considered a plausible reconstruction of Apoll onius’ method. The problem has stimulated much further work. Generalizations to three dimensions and beyond have been studied. The configuration of three mutually tangent circles has received particular attention. The Apollonian gasket is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy–Littlewood circle method. A solution to the problem may be either internally or externally tangent. An external tangency is one where the two circles lie on the same side of the tangent line, and one encloses the other in this case. By contrast, an internal tangent is one in which the circles lie in the same way at their point of contact, and in the other case, they lie on opposite sides of each other. The same holds for a line and a circle. Two distinct lines cannot be tangent in the plane, although two parallel lines can be considered as tangent at a point at infinity, although this is not considered true for two parallel circles. A circle may be internally tangent or externally inversive inversive geometry. The solution may be the given circle or a circle that may be externallyTangent to each of the given circles in an internal or external way, and the internal tangency of the circles is the sum of their radii between their centers.

The solution to this problem is known as the Apollonian gasket, after the Greek mathematician who solved it in the 4th century AD. It is used in navigation and positioning systems such as LORAN. It can also be used to find a position from the differences of its distances to three known points, such as in navigation systems like LORan. It has been used to solve the problem for the first time by Isaac Newton in Newton’s theory of relativity. It was also used by Joseph Diaz Gergonne to provide an elegant straightedge and compass solution, while other mathematicians used geometrical transformations such as reflection in a circle to simplify the configuration of theGiven circles. The general statement of Apollsonius’ problem is to construct one or more circles that were tangent to three given objects in a plane, where an object may be a line, a point or acircle of any size. These objects may be arranged in any way and may cross one another; however, they are usually taken to be distinct, meaning that they do not coincide. In practice, two distinct circles are said to intersect if they have a point in common. By definition, a points is tangent if it intersects them, that is, if it lies on them. If the angle between lines or circles at an intersection point is zero, they’re said to be tangents.

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