1. Proof of the property Maths Project The angle at the centre equals to 2 times the angle at the circumference Done by: JH306 Group D

2. Draw one diameter of the circle which passes through O, P, and intersects with the circle at point C Proof: Assume ∠APB=α ∠APO=β Hence ∠OPB=∠APB+∠APO =α+β Since OB=OP ∠OBP=∠OPB=α+β Therefore ∠BOP=180°-2∠OPB =180°-2(α+β)

3. Since OA=OP ∠AOP=180°-2∠OPA =180°-2β Hence ∠AOB=∠AOP-∠BOP =180°-2β -[180°-2（α+β）] =180°-2β-180°+2（α+β） =-2β+2α+2β =2α In conclusion, ∠AOB=2α=2∠APB