1. P
α
That is, no matter where
you place point P, the
A B
O
angle α is always 900
Note: AB is the diameter of the
circle whose centre is at O

2. P
x y Mark in the radius OP
2 isosceles triangles are thus formed
x y
A B
O So we can mark in angles x and y
Now we add the angles in Triangle APB
The angle sum is x + x + y + y = 2x + 2y
So 2x + 2y = 1800 x + y = 900
So angle APB is always 900

3. P
That is, no matter where you
place point P, the angle α = 2
O
always
B AB is a chord of the circle with
A centre O

4. P
Mark in the radius OP
z y 3 isosceles triangles are thus formed
So we can mark in angles x, y and z
O
y Angle APB = y + z
z
x Angle AOB = 1800 - 2x
x B
A
But, looking at triangle APB,
2x + 2y + 2z = 1800
1800 – 2x = 2(y + z)
Angle AOB = 2 x Angle APB

5. That is, no matter where you
P1 place point P, the angle APB
always the same – that is
P2
=
AB is a chord of the circle, which
B splits the circle into 2 segments
A

6. P2
Mark in the centre O, and form
P1 the triangle AOB
Let the angle AOB be 2
O
Thus angle AP1B will be
angle at centre = double angle at circumference)
B
A
Angle AP2B will be for the same reason
Thus angle AP1B = Angle AP2B

7. C
That is, if AB = CD, then d1 = d2
d2 D
O
d1
AB and CD are chords of equal length
B
A

8. C P is the mid-point of AB, and Q is the
Q mid-point of CD
D AP = CQ
O Mark in OA and OC (both are radii)
Triangle OPA is congruent to triangle
B OQC
P
A
OP = OQ