How do we prove that the angle at the centre is twice the angle at the

circumference?

OP

B

C

A

Y

With the given

diagram, we draw a

line OP and a line OC,

where AC is the

diameter of the

circle.

Since,

OP

B

C

A

Y

We know that AO =

OC = OP = OB,

because the radii of

a circle are all equal.

We can then

identify all the

isosceles triangles.

Also,

Now, we are ready to solve!

O

B

C

A

Y

Let’s look at AOB.

AOB lies on the

diameter of the circle.

Therefore, we can say

that AOB = 180- ( COP

+ POB).

We call that equation 1.

1: AOB = 180 – ( COP + POB)

P

O

B

C

A

Y

P

1: AOB = 180 – ( COP + POB)2: POB = 180 - 2 OPB

Now let’s look at angle POB.Angle POB is part of an isosceles triangle POB, where OPB = OBP .Therefore, we can say thatPOB = 180 – (OPB+ OBP)Or POB = 180 - 2 OPB

1: AOB = 180 – ( COP + POB)2: POB = 180 - 2 OPB

Given the 2 equations, we’ll substitute equation 2 into equation 1.

ÐAOB = 180 – ( COP + 180 - 2 OPB) = 180 – 180 - COP + 2 OPB = 2 OPB - COP

And… TADA!!!We have our 3rd equation formed!

O

B

C

A

Y

P

Getting back to our circle,

We know that

OPB = OPA + APB

Simple. No explanation

needed:D

And that will be equation

4!

3: AOB = 2 OPB - COP 4: OPB = OPA + APB

3: AOB = 2 OPB - COP 4: OPB = OPA + APB

Of course then, the next step will be to substitute 4 into 3!

AOB = 2 (OPA + APB) - COP = 2 OPA + 2 APB - COP 2 OPA + 2 APB - COP + AOB = 0

And here, we have our equation 5!!!

5: 2 OPA + 2 APB - COP + AOB = 0

Alright, back to the circle.

O

B

C

A

Y

P

We know that AOP is an isoceles triangle where OAP = OPA

Therefore, we can say that AOP + 2 OPA = 180However, we also know that AOP + COP = 180 (Angles on a straight line)

Thus, we can see that 2 OPA = COP (Which will be equation 6!)

*This is also a property of triangles, where the sum of 2 interior opposite s = the exterior .5: 2 OPA + 2 APB - COP + AOB = 0

6: 2 OPA = COP

5: 2 OPA + 2 APB - COP + AOB = 06: 2 OPA = COP

Again, we substitute equation 6 into 5.

2 OPA + 2 APB - 2 OPA + AOB = 02 APB - AOB = 0

2 APB = AOB

And…..

VIOLA!!!!

We’ve proven that AOB is

twice APB, where AOB is an

at the centre, and APB is an

at the circumference.

Conclusion:

Angle at the centre is

twice angle at the

circumference.

O

B

C

A

Y

P

A simple mathematical proof brought

to you by

JH301 Math Group C

Credits:

Mr Christopher Cheng

Miss Maureen Ng