Dip, 98 bytes
HC½+'0ÏEIH{C'0ÏEI
Explanation:
# Implicit input
H # Push length of input
C { # Divide byAdd 21
½+C # AddDivide 0.5
by 2
' # Convert to int
0Ï # Get string back
E # Push prefixes of string
I # Push prefixes[int(len/2+0.5)]prefixes[a]
# Implicit print
This could probably be much improved.