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In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency : internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the ...
Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have. In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this ...
In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the -axis at rational points. For each rational number p / q {\displaystyle p/q} , expressed in lowest terms, there is a Ford circle whose center is at the point ( p / q , 1 / ( 2 q 2 ) ) {\displaystyle (p/q,1/(2q^{2}))} and whose ...
For any two circles in a plane, an external tangent is a line that is tangent to both circles but does not pass between them. There are two such external tangent lines for any two circles. Each such pair has a unique intersection point in the extended Euclidean plane. Monge's theorem states that the three such points given by the three pairs of ...
Four mutually tangent circles define six points of tangency, which can be grouped in three pairs of tangent points, each pair coming from two disjoint pairs of circles. The three lines through these three pairs of tangent points are concurrent, and the points of concurrency defined in this way from the inner and outer circles define two more ...
Thus, as the solution circle swells, the internally tangent given circles must swell in tandem, whereas the externally tangent given circles must shrink, to maintain their tangencies. Viète used this approach to shrink one of the given circles to a point, thus reducing the problem to a simpler, already solved case.
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 ...
Mutually tangent circles. Given three mutually tangent circles (black), there are in general two other circles mutually tangent to them (red).The construction of the Apollonian gasket starts with three circles , , and (black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency.