Proof of an inequality with entropy and mutual information. Entropy of a random variable (a) is (h) : $H(a) = h$. 
Mutual information of (a) and (b) is (3h/4) : $I(a;b) = 3h/4$.
Mutual information of (a) and (c) is (3h/4) : $I(a;c) = 3h/4$.
It is needed to prove that $I(b;c) > h/2$
I tried to do it by a formula about link between mutual information and entropy: $I(a;b) = H(b) - H(b|a) = H(a) - H(a|b)$ but without any success.
Could you help me to prove it please. Thank you in advance. 


$I(a;b) = \frac{3h}{4}$  and  $I(a;c) = \frac{3h}{4}$.
Hence,
$I(a;b;c) \ge \frac{h}{2}$.
So if $I(b;c) \ne I(a;b;c)$ then $I(b;c) > \frac{h}{2}$
else  $I(b;c) \ge \frac{h}{2}$.
