Circles of Apollonius
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' definition of a circle. A circle is usually defined as the set of points P at a given distance "r" (the circle's radius) from a given point (the circle's center). However, there are  other, equivalent definitions of a circle. Apollonius discovered that a circle could be defined as the set of points P that have a given "ratio" of distances "k" =  to two given points (labeled A and B in Figure 1). These two points are sometimes called the foci. Proof using vectors in Euclidean spaces. Let "d", "d" be non-equal positive real numbers. Let C be the internal division point of AB in the ratio "d" : "d" and D the external division point of AB in the same ratio, "d" : "d". Then, Therefore, the point P is on the circle  which has the diameter CD. Proof using the angle bisector theorem. First consider the point formula_10 on the line segment between formula_11 and formula_12, satisfying the ratio. By the definition formula_13 and from the angle bisector theorem the angles formula_14 and formula_15 are equal. Next take the other point formula_16 on the extended line formula_17 that satisfies the ratio. So formula_18 Also take some other point formula_19 anywhere on the extended line formula_20. Also by the Angle bisector theorem the line formula_21 bisects the exterior angle formula_22. Hence, formula_23 and formula_24 are equal and formula_25. Hence by Thales's theorem  formula_26 lies on the circle which has formula_27 as a diameter. Apollonius pursuit problem. The Apollonius pursuit problem is one of finding whether a ship leaving from one point A at speed "v" will intercept another ship leaving a different point B at speed "v". The minimum time in interception of the two ships is calculated by means of straight-line paths. If the ships' speeds are held constant, their speed ratio is defined by μ. If both ships collide or meet at a future point, "I", then the distances of each are related by the equation: Squaring both sides , we obtain: Expanding: Further expansion: Bringing to the left-hand side: Factoring: Dividing by formula_37 : Completing the square: Bring non-squared terms to the right-hand side: Then: Therefore, the point must lie on a circle as defined by Apollonius, with their starting points as the foci. Circles sharing a radical axis. The circles defined by the Apollonian pursuit problem for the same two points A and B, but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a "hyperbolic  pencil". Another family of circles, the circles that pass through both A and B, are also called a pencil, or more specifically an "elliptic pencil". These two pencils of Apollonian circles intersect each other at right angles and form the basis of the bipolar coordinate system. Within each pencil, any two circles have the same radical axis; the two radical axes of the two pencils are perpendicular, and the centers of the circles from one pencil lie on the radical axis of the other pencil. Solutions to Apollonius' problem. In Euclidean plane geometry, Apollonius's problem is to construct circles  that are tangent to three given circles in a plane. Three given circles generically have eight different circles that are tangent to them and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed. Apollonian gasket. By solving Apollonius' problem repeatedly to find the inscribed circle, the interstices between mutually tangential circles can be filled arbitrarily finely, forming an Apollonian gasket, also known as a "Leibniz packing" or an "Apollonian packing". This gasket is a fractal, being self-similar and having a dimension "d  that is not known exactly but is roughly 1.3, which is higher than that of a regular (or rectifiable) curve (d" = 1) but less than that of a plane ("d" = 2). The Apollonian gasket was first described by Gottfried Leibniz in the 17th century, and is a curved precursor of the 20th-century Sierpiński triangle. The Apollonian gasket also has deep connections to other fields of mathematics; for example, it is the limit set of Kleinian groups; see also Circle packing theorem. Isodynamic points of a triangle. The circles of Apollonius may also denote three special circles formula_42  defined by an arbitrary triangle formula_43. The circle formula_44 is defined as the unique circle passing through the triangle vertex formula_45 that maintains a constant ratio of distances to the other two vertices formula_46 and formula_47 (cf. Apollonius' definition of the circle above). Similarly, the circle formula_48 is defined as the unique circle passing through the triangle vertex formula_46 that maintains a constant ratio of distances to the other two vertices formula_45 and formula_47, and so on for the circle formula_52. All three circles intersect the circumcircle of the triangle orthogonally. All three circles pass through two points, which are  known as the isodynamic points formula_53 and formula_54 of the triangle. The line connecting these common intersection points is the radical axis for all three circles. The two isodynamic points are inverses of each other relative to the circumcircle of the triangle. The centers of these three circles fall on a single line (the Lemoine line). This line is perpendicular to the radical axis, which is the line determined by the isodynamic points.