If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of
these numbers must be 1. Now for the complexity! When are the other
numbers real and when are they complex?

This problem in geometry has been solved in no less than EIGHT ways
by a pair of students. How would you solve it? How many of their
solutions can you follow? How are they the same or different? Which
do you like best?

Thousand Words

Age 16 to 18 Challenge Level:

Care needs to be taken over values of $\alpha$ and $\beta$ greater
than $2\pi$.