Fuhrmann's Theorem
FuhrmannsTheorem
Let the opposite sides of a convex cyclic hexagon be a, a^', b, b^', c, and c^', and let the polygon diagonals e, f, and g be so chosen that a, a^', and e have no common polygon 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 TheoremExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Fuhrmann's TheoremCite this as:
Weisstein, Eric W. "Fuhrmann's Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FuhrmannsTheorem.html