Shoe buckle - Agatha Christie

The dentist was found dead with a gun thrown to the ground next to his right hand, and then one of his patients was found dead from a massive dose of the doctor's anesthetic. It is a clear context for a murder-suicide. But why would a doctor kill his patient in the midst of a busy day with appointments?!

shoe buckle holds key to solution; Will Poirot succeed in putting together the clues and solving the mystery?