Puzzle #161: For the Birds

You have some birds: chickens, turkeys, and ducks. There are twice as many chickens as turkeys. The number of ducks is six more than the average of the number of chickens and the number of turkeys. If you have fewer than 50 birds, what is the maximum number of birds you could have?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, July 6, 2016 at noon Pacific Time. I will post the answer shortly after.