If a,b,c are in Harmonic Progression,

That means that 1/a, 1/b, 1/c are in arithmetic progression, i.e.








We are to show that

(1)   


All terms are of the form 

Multiply top and bottom by x(y+z)



So (1) becomes

(2)    

(2) will be true if and only if

 

which will be true if and only if



which will be true if and only if 



which will be true if and only if



Substitute 



which will be true if and only if



Which will be true if and only if 



And since this is true, the problem is proved.

Edwin