Fuhrmann's theorem

Students will learn what Fuhrmann's theorem is, and why it's true.

Theorem:

Suppose that ABCDEF is a convex cyclic hexagon.
Then

|AD||BE||CF| = |AB||CD||EF|
             + |AF||BC||DE|
             + |AB||DE||CF|
             + |BC||EF||AD|
             + |CD||AF||BE|

Proof:

This is obtained by applying ptolemy's theorem to the
convex cyclic quadrilaterals ABDE, BCDF, ADEF, ABEF

|AD||BE| = |AB||DE|+|AE||BD|
|BD||CF| = |BC||DF|+|BF||CD|
|AE||DF| = |AD||EF|+|AF||DE|
|AE||BF| = |AB||EF|+|AF||BE|

Multiplying the first by |CF|, we get

|AD||BE||CF| = |AB||DE||CF|+|AE||BD||CF|

By the second, we can replace |BD||CF| by
|BC||DF|+|BF||CD| in the second factor on the right

|AD||BE||CF| = |AB||DE||CF|+|AE||BC||DF|+|AE||BF||CD|

Using the third and fourth, we can replace |AE||DF| by
|AD||EF|+|AF||DE|, and |AE||BF| by |AB||EF|+|AF||BE|.
This gives the stated result.