In To Kill a Mockingbird, what does Atticus show in his cross examination of Sheriff Tate, and what are the main points in Tate's evidence?

As explained in the previous answer, we learn from Tate's evidence that the nature of Mayella's injuries would appear to suggest that she was attacked by someone who mostly used his left hand, and that he also put both hands around her throat. In the light of this evidence, the fact that Bob Ewell, Mayella's father, is left-handed takes on great significance. Atticus emphasizes Bob's left-handedness in court, although Bob, who appears decidedly dim-witted at this point, fails to see the implications.

Jem is excited; he exults that 'we've got him' (referring to Bob Ewell). However Scout at this stage is more sceptical; she thinks Tom might be left-handed too and that 'Jem was counting his chickens'. But then when Tom's cross-examination begins, she sees him clearly for the first time and realizes that he has a badly crippled left arm - as the result of catching it in some machinery when he was young. To any impartial jury, this would rule out Tom Robinson as a suspect. Of course, the point is that it is not an impartial jury, but rather one blighted by racial prejudice who goes on to convict Tom simply because he is black. Had he not been black, he likely would not even have been charged.

Tate's evidence on the nature of Mayella's injuries, then, goes right to the heart of the matter, and appears to point to Mayella's own father Bob as being her attacker; yet this evidence still proves insufficient to acquit Tom Robinson.