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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 ...
If = + is the distance from c 1 to c 2 we can normalize by =, =, = to simplify equation (1), resulting in the following system of equations: + =, + =; solve these to get two solutions (k = ±1) for the two external tangent lines: = = + = (+) Geometrically this corresponds to computing the angle formed by the tangent lines and the line of ...
In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that the radius of a circle is always normal to the circle itself. With this in mind Descartes would ...
If , are tangent from different sides of (one in and one out), is the length of the interior common tangent. The converse of Casey's theorem is also true. [ 4 ] That is, if equality holds, the circles are tangent to a common circle.
Monge's theorem states that the three such points given by the three pairs of circles always lie in a straight line. In the case of two of the circles being of equal size, the two external tangent lines are parallel. In this case Monge's theorem asserts that the other two intersection points must lie on a line parallel to those two external ...
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.
It consists of a family of real circles, all tangent to each other at a single common point. The degenerate circle with radius zero at that point also belongs to the pencil. A family of concentric circles centered at a common center (may be considered a special case of a hyperbolic pencil where the other point is the point at infinity).
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. The word "tangent" comes from the Latin tangere, "to touch".