What is atticus tryng to prove in the court proceedings when he has the sheriff testify about which side of the face Mayella was injured on?
When Atticus called the sheriff, Heck Tate, he establishes for the jury the injuries that Mayella suffered in the attack. She has bruises all around her neck and on the right side of her face. These injuries show that she was strangled by two hands and beaten on the right side of her face by a person who leads with his left.
Atticus gets Bob Ewell to confirm the testimony of Heck Tate. He doesn't dispute anything the sheriff says. Then Atticus gets Heck Tate to sign his name. Although Bob thinks Atticus is doing this to prove he can't write, he's really doing it to show that Bob is left-handed.
When Tom Robinson testifies later in the trial (the same night), it is apparent to everyone, including the jury, that Tom could not be guilty of attacking Mayella in this way. His left arm is completely useless. It hangs weakly by his side because all the muscles were torn from the bone in a cotton gin accident years ago.
Atticus has successfully proven in the minds of the jury and spectators that Tom is not guilty. Although he doesn't win the case, Atticus has revealed the truth.