Prove that the equal-length concatenation of regular languages is context free
To expand a bit on @babou's comment, let $M_A$ and $M_B$ be the FAs for $A, B$ respectively. Make a PDA that acts like this:

(1) repeat
      read the input, using MAs transition rules.
      For each character read, push a marker on the stack.
    until MA enters a final state or no input remains
(2) if no input remains and MA is in a final state, go to (3)
    else if no input remains and MA is in a non-final state, halt and reject
    else nondeterministically go to (1) and (3) [guessing that we're done with the first half]
(3) repeat
      run the PDA on any remaining input, using MBs transition rules
      For each character read, pop the stack
    until the stack is empty or no input remains
(4) if MB is in an accept state, no input remains, and the stack is empty, halt and accept
    else halt and reject


(I apologize in advance for any missing details/programming errors, but the intent, I hope, is clear: process the first part of the input, guess that we're in the middle and process the second part.)