It ain't elegant but it works:

1) if

or

2) if

or

3) if

Now:

3) is the easiest case. and are contradictory. Thus .

So consider case 1):

1) if

We know that and a is positive, so . This means that either both b, c are positive or both b, c are negative.

If both b, c are positive then is trivial since the RHS is negative, and we have no contradiction.

If both b, c are negative then

Now if then

is a contradiction because and the RHS is negative.

If and we know that c is negative then cannot be true because , a contradiction.

Thus we are left with .

Thus

another contradiction.

Thus both b and c cannot be negative.

Thus case 1) says that if a is positive, so are both b and c.

Case 2) works in a similar fashion to show that, at best, all three of a, b, and c must be negative. (I leave this for you to show.) This again contradicts .

-Dan