cocircularly

Cocircularity refers to a geometric property where multiple circles share the same center point. In a set of cocircular circles, all circles lie on the same plane and are concentric. This means that the distance from the common center to any point on any of the circles is constant for each individual circle, but different between the circles. For example, if you have a bullseye target, the rings of the target are cocircular because they all have the same center.

The concept of cocircularity is fundamental in geometry and has applications in various fields. In astronomy, orbits of planets around a star can be approximated as cocircular under certain conditions, though in reality they are typically elliptical. In engineering, the design of gears, bearings, and rotating machinery often involves cocircular components to ensure smooth and efficient operation. In computer graphics and design, creating concentric patterns or layouts relies on the principle of cocircularity.

Mathematically, a collection of circles with equations $(x-h)^2 + (y-k)^2 = r_1^2$, $(x-h)^2 + (y-k)^2 = r_2^2$, ..., $(x-h)^2 + (y-k)^2 = r_n^2$ are cocircular if they all share the same center $(h, k)$ but have different radii $r_1, r_2, ..., r_n$. The property simplifies many geometric calculations and proofs involving circles.