students must be able to understand why ∠ at Centre = 2 times ∠ at Circumference.
Proof:
Let ∠AOC = 2a
Let ∠BOC = 2b
Then ∠AOB = 360° - 2a – 2b
∠ OCA = 90° – a (isosceles triangle)
∠BCO = 90° – b (isosceles triangle)
Therefore, ∠ACB = (90° – a) + (90° – b) = 180° – a – b
Hence, ∠AOB = 2∠ACB (∠ at Centre = 2 times ∠ at Circumference) Proven
http://weelookang.blogspot.sg/2014/12/geogebra-angle-at-centre-equal-twice.html

angle at centre equal twice angle at circumference

Steps:
1. Compare angles at the centre of a circle with angle touching the circumference.
2. vary the ∠ at Centre O for which it is acute less than 90 °
3. write down the value of ∠ at Centre O and ∠ at Circumference point A
4. vary the ∠ at Centre O for which it is obtuse more than 90° and less than 180°.
5. do step 3
6. vary the ∠ at Centre O for which it is reflex more than 180°.
7. do step 3
Thinking:
looking at the evidence of the table of recorded values, suggest a relationship between
∠ at Centre O and ∠ at Circumference point A.
Conclusion:
∠ at Centre = 2 times ∠ at Circumference.