1. Proofs for circle theorems
Tuesday, 13 July 2010

2. 1. Angle subtended at the centre
The angle subtended at the centre from an arc is double the angle at the circumference.  y x
180 – 2x C x B
180 – 2y
2x+2y
= 2(x+y)  y A

3. 2. Angles subtended from the same arc.
Angles subtended from the same arc are equal.   C B A

4. 3. Angles in a semi-circle.
The largest angle in a semi-circle will always be 90 90 C A B

5. 4. The Angle between a Tangent and its radius.
Definition:
A tangent is a line that will touch the circle at one point only. (i.e. it does not cut the circle) C
Tangent
Assume that the tangent is not perpendicular to the radius. Now there must be a perpendicular from the center of the circle to the tangent. It must intersect the tangent elsewhere (apart from the point on intersection). Now since a perpendicular is the shortest distance of a point from a line, the perpendicular must have distance < radius of the circle (since line from point of intersection of tangent to center of circle is a radius). => the foot of the perpendicular must lie within the circle. =>the line meets the circle elsewhere and the line is not really a tangent! Thus there is a contradiction. Therefore the radius is the shortest distance. Consequently, it must be perpendicular to the tangent at the point of intersection 90 A
The angle between a tangent an its radius will always be 90

6. 5. Angles in a cyclic quadrilateral.
Definition:
A cyclic quadrilateral is any four-sided polygon whose four corners touch the circumference of the circle
180 –  d b 2
360 –2
Opposite angles in a cyclic quadrilateral add up to 180  c

7. 4. The Angle between a Tangent and a chord.
Definition: A chord is any straight line which touches the circumference at two points. The largest chord possible is called the diameter.
=180 – 90 – (90 – ) =  
Tangent 90
90 – 
Chord
The angle between a tangent a chord is equal to the angle in the alternate segment.