SOLUTION: if a, b, c are in hp show that 1/a + 1/b+c, 1/b + 1/(c+a), 1/c + 1/(a+b) are also in hp.

Proposition to prove:

Two things we must know

1.  Three terms p,q,r are in hp if and only if their reciprocals
    1/p, 1/q, 1/r are in ap.

2.  Three terms u,v,w are in ap if and only if v-u = w-v

That is true if and only if

1/a, 1/b, 1/c are in ap.

That is true if and only if

(1)       

That will be true true if and only if after getting LCD's, we can show that:

, , and 

are in hp.  That will be true if and only if

, , and 

are in ap.

They have the same denominators so they will be in ap if and only if their
numerators are in ap.  That will be true if and only if



That will be true if and only if



That will be true if and only if



That will be true if and only if we can divide through by abc
and this will be true:



That will be true if and only if we can cancel and get



That will be true if and only if we multiply through by -1, and get



That will be true if and only if we can reverse the terms and get



And we have that that is true in equation (1) above.  So the proposition is
proved.

Edwin