Circle Theorems
Same segment theorem
Angles subtended by the same segment (made by a chord and an arc) at the circumference are equal.
Centre-Circumference Theorem
One of the rules of geometry is that the angle subtended at the centre of a circle is double the size of the angle subtended at the edge from the same two points.
You will often see the word 'subtended' used in maths questions. Do not be put off by it. 'Subtended' is just a technical way of saying 'made' or 'hanging down'.
Cyclic Quadrilateral Theorem
A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180º.
In the cyclic quadrilateral below, angles A + C = 180º, and angles B + D = 180º.
Angles in a Semi-Circle
We have just seen that the angle at the centre of a circle is double the size of the angle at the circumference.
The angle at the centre (AOB) is twice the angle at the circumference (APB). As AOB is 180°, it follows that APB is 90°.
AOB is the diameter, so it follows that the angle in a semicircle is always a right angle: and that angles in the same segment are equal.
