__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 90*^{o} and the angle in a semicircle is 90^{o}

*This is because the 'angle' at the
centre is 180*^{o}

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 180^{o}

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 180^{o},
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...