How do you prove to your best friend that she's not fat?

Try showing her people who are fatter than her, but more confident. It may also help to calculate where she falls on the BMI chart and show her where she falls. This can be tricky sometimes because the BMI charts don't account for body type; I'm very athletic and the BMI says I'm obese, but I am obviously not. There are also other measurements such as waste size that can be used to judge if someone is overweight.