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Circle theorems

 The angle α is subtended by the arc or chord AB. See Geometry for explanations of these terms.

Any triangle drawn by joining the centre to two points on the circumpherence is isosceles because two of the sides are radii so angle a = angle b

 

Theorem 1

Angles subtended by the same chord are equal provided that they are in the same segment

If we go to the other segment, the angle is 180 minus the one in the original segment (see cyclic quadrilaterals later)

e.g. 55 + 125 = 180

 

Theorem 2

The two angles below are subtended by the same arc. The one at the centre is twice the one at the circumpherence

Because of this, the angle subtended on a diameter is 90o and the angle in a semicircle is 90o

This is because the 'angle' at the centre is 180o

N.B. Although it is easy to remember this rule as the 'arrowhead' picture, be careful since in some cases the shape may not always look line an arrowhead

 

Proof for circle theorem 2

1)    Triangle ZOX is isosceles

2)    Angle XOZ is 180 - 2a

3)    Its external angle (angle XOY) is therefore 180 - (180 - 2a), which is 2a

4)    We can work out the same for the other side and find that if angle YZO is b then angle YOW is 2b.

5)    Hence angle XOY is 2a + 2b = 2(a+b). Angle XZY is a+b. This is twice XOY hence the circle theorem works!

 

Cyclic quadrilaterals

A cyclic quadrilateral is one that has all its corners touching the circumpherence of a circle

 

Cyclic quadrilateral theorem 1

Opposite angles in a cyclic quadrilateral add up to 180o

 

The proof of this...

 

Starting from a, using circle theorem 2, we can find 2a. Hence the internal angle at the centre is360 - 2a and so the opposite angle to a in the cyclic quadrilateral is half of this, which is 180 - a.

 

Cyclic quadrilateral theorem 2

Because of theorem 1, that opposite angles add up to 180o, the exterior angle is equal to the opposite interior angle

 

Tangent theorems

Tangent theorem 1

A Radius drawn to the point of contact of a tangent is always perpendicular to it

 

Tangent theorem 2

Triangles TPO and TQO are congruent (identical)

 

Tangent theorem 3 - The alternate segment theorem

These two angles are equal

The proof of this...

Starting at the tangent, x + the angle next to it must be 90 due to tangent theorem 1, hence the 'angle next to it' must be 90 - x

The red triangle is isosceles so the other 90 - x can be marked in

Angles in a triangle add up to 180 so 2x can be marked in

Using circle theorem 2, the angle at the circumpherence must be x

SO IT WORKS!

 

 

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