Descartes' theorem is most easily stated in terms of the circles' curvatures. The signed curvature (or bend) of a circle is defined as k = ± 1 / r {\displaystyle k=\pm 1/r} , where r {\displaysty… See more

To determine a circle completely, not only its radius (or curvature), but also its center must be … See more

Three congruent circles
When three of the four circles are congruent, their centers form an equilateral triangle, as do their points of tangency. The two possibil… See more

Any triangle in the plane has three externally tangent circles centered at its vertices. Letting A , B , C {\displaystyle A,B,C} be the three points, a , b , c {\displaysty… See more

Arbitrary four-circle configurations
Descartes' theorem can be expressed as a matrix equation and then generalized to other configurations of four oriented circles by changin… See more