The circle theorems can be proved using knowledge of basic angle facts and properties of 2D shapes. If you are asked to prove a circle theorem, adding in any radii and finding all equal angles will help. There are two types of proofs that can be used for the different circle theorems.
In the diagram below, , and are points on the circumference of a circle, centre . is a tangent to the circle.
Prove that angle and angle are equal.
Begin by joining the point O to the point C and continue the line through to the circumference so that a diameter is drawn on the diagram.
Label this new point on the circumference F.
Join the point F to the point B on the circumference.
Angle CBF = 90° The angle in a semicircle is always 90°
The line OC is a radius, so it will meet the tangent DE at 90°. Let angle .
Angle The radius meets a tangent at 90°
Angles in a triangle add up to 180°, use this to find the angle CFB in terms of .
The angles in a triangle add up to 180°
Angles subtended at the circumference from the same arc are equal. Use this to find an expression for angle BAC in terms of .
We have already stated that .
Therefore Angle BCE = Angle BAC