fuhrmann's theorem in euclidean geometry
Suppose that ABCDEF is a convex cyclc hexagon.
Then ADBECF =
ABCDEF+AFBCDE+ABDECF+
BCEFAD+CDAFBE.
proof
This is obtained by applying ptolemy's theorem to the
convex cyclic quadrilaterals ABDE, BCDF, ADEF, ABEF
ADBE = ABDE+AEBD,
BDCF = BCDF+BFCD,
AEDF = ADEF+AFDE,
AEBF = ABEF+AFBE.
Multiplying the first by CF, we get
ADBECF = ABDECF+AEBDCF.
By the second, we can replace BDCF by
BCDF+BFCD in the second factor on the right :
ADBECF = ABDECF+AEBCDF+AEBFCD.
Using the third and fourth, we can replace AEDF by
ADEF+AFDE, and AEBF by ABEF+AFBE.
This gives the stated result.

