A is better than C
B is not better than C
Therefore A must be better than B.
Doesn't feel quite right, does it?
Now try this one:
A is significantly better than C
B is not significantly bettter than C
Therefore A must be significantly better than B
Confidence interval on A excludes C
Confidence interval on B does not exclude C
Therefore A's confdience interval must exclude B
Hopefully you can now see clearly what's been wrong all along.
This failure of logic can be disguised statistical testing: A vs C, B vs C. But the only way you can be sure is to do the direct comparison: A vs B. The golden rule in statistical testing is to use the direct, appropriate test for the thing you are interested in, not some other indirect comparison. Insert the names of a few common drugs instead of A, B, C and you can see why this is hugely important.
This and more in a compendium of Ben Goldacre's writings:
An excellent way to fill those long days between Christmas and New Year!