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These equations express that the tangent line, which is parallel to , is perpendicular to the radii, and that the tangent points lie on their respective circles. These are four quadratic equations in two two-dimensional vector variables, and in general position will have four pairs of solutions.
Tangent lines to circles; Circle packing theorem, the result that every planar graph may be realized by a system of tangent circles; Hexafoil, the shape formed by a ring of six tangent circles; Feuerbach's theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles; Descartes' theorem; Ford circle; Bankoff circle
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.
An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2] The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.
When the centre of the circle is at the origin, then the equation of the tangent line becomes + =, and its slope is =. Properties The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality ).
Constructing a tangent using Thales's theorem. Thales's theorem can be used to construct the tangent to a given circle that passes through a given point. In the figure at right, given circle k with centre O and the point P outside k, bisect OP at H and draw the circle of radius OH with centre H.
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.
The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then [12] ... Two pairs of tangent circles. Above internally and below externally tangent.