How did Dr. Haledjian know who murdered Jim in the case of the Big Deal?

Dr. Haledjian deduced that the murderer of Jim in the case of the Big Deal was Alan Hayes based on the fact that the gun used in the murder was fired by someone left-handed, and Jim's wife mentioned that Alan was left-handed. Additionally, Alan's relationship with Jim's wife and his knowledge of the car tire tracks at the scene further implicated him in the crime.