Master all circle theorems for GCSE Maths with clear diagrams, explanations, and exam tips. Download as PDF for offline revision.
The angle at the centre is twice the angle at the circumference when subtended by the same arc.
When two points on a circle are connected to both the centre and a point on the circumference, the angle at the centre is always double the angle at the circumference.
Angles subtended by the same arc at the circumference are equal.
Any angles drawn from the same chord to points on the same side of the circumference will always be equal.
The angle in a semicircle is always 90° (a right angle).
When a triangle is drawn inside a semicircle with the diameter as its base, the angle at the circumference is always 90°. This is also known as Thales' theorem.
Opposite angles in a cyclic quadrilateral add up to 180°.
A cyclic quadrilateral has all four vertices on the circumference of a circle. The opposite angles always sum to 180°.
A tangent to a circle is perpendicular to the radius at the point of contact.
Where a tangent touches the circle, it meets the radius at exactly 90°. The radius is drawn from the centre to the point where the tangent touches.
Two tangents drawn from the same external point are equal in length.
If you draw two tangent lines from a point outside the circle, both tangent segments will have exactly the same length.
The angle between a tangent and a chord equals the angle in the alternate segment.
The angle formed between a tangent and a chord at the point of contact is equal to the inscribed angle subtended by the chord on the opposite side.
A perpendicular line from the centre of a circle to a chord bisects the chord.
If you draw a line from the centre perpendicular to a chord, it will cut the chord exactly in half. This works the other way too - a line from the centre to the midpoint of a chord is perpendicular to it.
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