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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 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 ...
They are all named for Frederick Soddy, who rediscovered Descartes' theorem on the radii of mutually tangent quadruples of circles. Any triangle has three externally tangent circles centered at its vertices. Two more circles, its Soddy circles, are tangent to the three circles centered at the vertices; their centers are called Soddy centers.
For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center. Both ...
The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different ...
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.
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 ...
Draw circle C that has PQ as diameter. Draw one of the tangents from G to circle C. point A is where the tangent and the circle touch. Draw circle D with center G through A. Circle D cuts line l at the points T1 and T2. One of the required circles is the circle through P, Q and T1. The other circle is the circle through P, Q and T2.