The document proves that in a circle with lines OA, OB, and OP subtending from the center, angle AOB is twice angle APB. Angle OBP is shown to equal the sum of angles OPA and APB. The exterior angle theorem is then applied to show angle ACB is the sum of angles APB and OBP, and angle AOC is equal to angle ACB minus angle OPA, demonstrating angle AOB is twice angle APB.