What does Atticus show in his cross-examination of Sheriff Tate in To Kill a Mockingbird?

In his cross-examination of Sheriff Tate, Atticus establishes two things. First, that the bruises and wounds on Mayella's face after the incident were on the right side of her face. Sheriff Tate then volunteers that she had bruises elsewhere, especially around her throat, as if someone had grabbed her around the neck and choked her. Later, we see the significance of these facts. As it turns out, Tom Robinson's left arm is essentially crippled, shriveled up and largely useless. Because a blow to the right side of Mayella's face would have likely been delivered by someone's left hand and because it was very unlikely that Tom, with his crippled left arm, could have put both hands around Mayella's neck to choke her, Atticus begins to establish a more-than-reasonable doubt that Tom was responsible for the crime. As it turns out, of course, these facts don't really matter. Tom, a black man, is convicted by the all-white jury.