Directed Angle

The symbol $\dirang ABC$ denotes the directed angle from $AB$ to $BC$, which is the signed angle through which $AB$ must be rotated about $B$ to coincide with $BC$. Four points $ABCD$ lie on a Circle (i.e., are Concyclic) Iff $\dirang ABC=\dirang ADC$. It is also true that

$$\dirang l_1l_2+\dirang l_2l_1=0^\circ {\rm\ or\ } 360^\circ.$$

Three points $A$, $B$, and $C$ are Collinear Iff $\dirang ABC=0^\circ$ or $180^\circ$. For any four points, $A$, $B$, $C$, and $D$,

$$\dirang ABC+\dirang CDA=\dirang BAD+\dirang DCB.$$

See also Angle, Collinear, Concyclic, Miquel Equation

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 11-15, 1929.