Circle Theorems

1
Circle Theorems

• The theorems

2
Some given facts
Isosceles Triangle
The Tangent is perpendicular to the radius
3
First a triangle using a diameter
Two minute challenge ? ABC 900, prove it!
B
x
y
A
x
Make two isosceles triangles Label the equal
angles of one and express the third using this
y
C
180 2x
Do the same for the other isosceles triangle
180 2y
We know that the angles of ?ABC 1800 So x x
y y 1800
So x y 900 QED
4
Second two tangents
Examine this diagram and Decide what can be
proved here
B
x
180 2x
A
x
O
x
Two minute challenge Show that ?AOB ? AOC
x
180 2x
C
5
Third double angle rule
B
180 2y
Five minute challenge Show that ?AOC twice ? ABC
180 2x
y
x
y
O
C
x
360 (180 2x) (180 2y)
A
360 180 2x 180 2y
2x 2y
twice ? ABC QED
6
Fact the double angle rule works both ways
B
?AOC twice ? ABC
O
C
and
A
Reflex ?AOC twice ? ADC
D
7
Fourth equal angles from a chord
B
Five minute challenge Show that ?ABC ? ADC
D
ABC half AOC
O
And
ADC half AOC
A
C
So they are equal QED
Note that the made angles above chord AC are
equal And the angles made below chord AC are
equal, but not to those above
8
Another fact
The Chord is perpendicular to the radius that
bisects it
Since its parallel to the diameter and Tangent
shown
9
Fifth cyclic quadrilateral
Five minute challenge Show that ?ABC ? ADC
1800
B
Now what?
y
2x 2y 360
2x
O
So
C
2y
A
x y 180
QED
x
D
10
Sixth Alternate segment theorem
2x 2y
½ (180 2x 2y)
90 x y
A
Five minute challenge Show that ?ACD ? ABC
B
x
y
O
Now add up the angles that make ABC
x
y
ABC 90 x y y
C
90 - x
D
? ACD QED
90 x
11
Now try and draw each of the angle facts you know
about circles. Without looking!
You want to be able to sketch ten facts from
memory. e.g.