Casey's Theorem

Four Circles are Tangent to a fifth Circle or a straight Line Iff

$$t_{12}t_{34}\pm t_{13}t_{42}\pm t_{14}t_{23}=0,$$

where $t_{ij}$ is a common Tangent to Circles $i$ and $j$.

See also Purser's Theorem

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 121-127, 1929.