Fuhrmann's Theorem

Let the opposite sides of a convex Cyclic Hexagon be $a$, $a'$, $b$, $b'$, $c$, and $c'$, and let the Diagonals $e$, $f$, and $g$ be so chosen that $a$, $a'$, and $e$ have no common Vertex (and likewise for $b$, $b'$, and $f$), then

$$efg=aa'e+bb'f+cc'g+abc+a'b'c'.$$

This is an extension of Ptolemy's Theorem to the Hexagon.

See also Cyclic Hexagon, Hexagon, Ptolemy's Theorem

References

Fuhrmann, W. Synthetische Beweise Planimetrischer Sätze. Berlin, p. 61, 1890.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 65-66, 1929.