First, note that for any 3 Random variables (X,Y,Z):

Cov(X+Y,Z) = Cov(X,Z) + Cov(Y,Z)

Since

Apply that rule twice:

Cov(aA + (1-a)C,bB + (1-b)C) = Cov(aA,bB) + Cov(aA,(1-b)C) + Cov((1-a)C,bB) + Cov((1-a)C,(1-b)C)

...

You can evaluate each of those terms provided you know that Cov(aX,Y) = aCov(X,Y)

Once you have the covariance, you should be able to get the correlation.