Several sets of circles associated with Apollonius of Perga
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.
The main uses of this term are fivefold:
Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ratio of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2.
The Apollonian circles are two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis of bipolar coordinates.
The circles of Apollonius of a triangle are three circles, each of which passes through one vertex of the triangle and maintains a constant ratio of distances to the other two. The isodynamic points and Lemoine line of a triangle can be solved using these circles of Apollonius.
Apollonius' problem is to construct circles that are simultaneously tangent to three specified circles. The solutions to this problem are sometimes called the circles of Apollonius.
The Apollonian gasket—one of the first fractals ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
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plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apolloniusof Perga (c. 262...
plan from Apollonius himself, who visited Pergamon, Eudemus insisted Apollonius send him each book before release. At this stage Apollonius was likely...
up Apollonius in Wiktionary, the free dictionary. Apollonius (Ancient Greek: Απολλώνιος) is a masculine given name which may refer to: Apolloniusof Athens...
These circles form the basis for bipolar coordinates. They were discovered by Apolloniusof Perga, a renowned Greek geometer. The Apollonian circles are...
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problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction ofcircles that are tangent...
point on the polygon. An example of this is the mice problem. Logarithmic spiral Tractrix CirclesofApollonius#Apollonius pursuit problem Pursuit–evasion...
is equal to the sum of their radii Apollonius' problem is to construct circles that are tangent to three given circles. If a circle is iteratively inscribed...
intersection points of the three circlesofApollonius associated with triangle of a triangle A B C , {\displaystyle ABC,} the three circles that each pass...
intersection of the three lines, and the three exscribed circles. A general Apollonius problem can be transformed into the simpler problem ofcircle tangent...
{A-B}{2}}\end{array}}} Apollonius' theorem Apolloniusof Perga (262–190 BC), geometer and astronomer Apollonius problem Apollonian circles Isodynamic point of a triangle...
infinite radius. Systems of mutually tangent circles were studied by Apolloniusof Perga, after whom the problem ofApollonius and the Apollonian gasket...
Theon of Alexandria, Anthemius, and Eutocius. The following works are extant only in Arabic translations: Apollonius, Conics books V to VII Apollonius, Cutting...
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named after Greek mathematician Apolloniusof Perga. The construction of the Apollonian gasket starts with three circles C 1 {\displaystyle C_{1}} , C 2...
prove the following theorems/results: Coordinates of the incenter of a triangle CirclesofApollonius Alfred S. Posamentier: Advanced Euclidean Geometry:...
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