Answer: 

One of the readers of this website, John Navratil, emailed me to tell me how to solve this one.  The letter in the middle is the "average" of the six letters around the edge.  Change each letter into a number using A=1, B=2, etc.  Then average the outer letters.  On the left side, B+F+M+Y+O+E is 2+6+13+25+15+5=66, so their average is 11, which corresponds to K.  On the right side H+Q+A+J+U+C is 8+17+1+10+21+3=60, so their average is 10, which corresponds to J.