Subjects

Maths

Circles

Properties of Tangents

Proof: Segments Tangent to Circle from Outside Point Are Congruent

152

•

Dec 7, 2025

•

2 pages

The 7 Circle Theorems Explained with Examples

Libby

@libbysnotes

Circle theorems are essential geometric rules that help you solve... Show more

circle theorems
When a radius meets a
tangent, the angle will
always be 90°
2x
The angle subtended at
the centre of a cirle is
twice the ang

Basic Circle Theorems

Radius and tangent relationships form the foundation of circle geometry. When a radius meets a tangent at the point where it touches the circle, they always form a perfect 90° angle - this is one of the most reliable rules you'll use.

Here's something quite clever about tangents: from any point outside a circle, you can draw exactly two tangent lines. The distances from that external point to where each tangent touches the circle are always equal. This equal tangent lengths rule often helps you find missing measurements.

The angle at the centre theorem shows a beautiful relationship between angles. Any angle formed at the centre of a circle is exactly twice the size of the angle formed at the circumference when both angles use the same arc. This 2:1 ratio appears constantly in circle problems.

Quick tip: Remember "centre doubles circumference" - the angle at the centre is always double the angle at the circumference for the same arc.

Finally, any angle in a semicircle is always 90°. If you spot a triangle where one side is the diameter of a circle, you've found yourself a right-angled triangle.

Advanced Circle Theorems

Angles in the same segment are always equal to each other. This means if you have multiple angles that sit on the same side of a chord and use the same arc, they'll all have identical measurements - quite handy for solving complex problems.

Cyclic quadrilaterals $four-sided shapes inscribed in circles$ follow a special rule: opposite angles always add up to 180°. If you know three angles in a cyclic quadrilateral, you can instantly find the fourth one.

The alternate segment theorem is perhaps the trickiest but most elegant rule. The angle between a tangent and a chord equals the angle in the alternate segment. Think of it as angles "on opposite sides" being equal.

Exam strategy: These theorems often combine in questions, so look for multiple rules working together rather than just one isolated theorem.

Understanding these patterns means you'll recognise the theorems quickly in exam questions. Practice identifying which theorem applies first, then apply the rule confidently.

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Subjects

Maths

Circles

Properties of Tangents

Proof: Segments Tangent to Circle from Outside Point Are Congruent

Maths

•

152

•

Dec 7, 2025

•

2 pages

The 7 Circle Theorems Explained with Examples

Libby

@libbysnotes

Circle theorems are essential geometric rules that help you solve angle and length problems involving circles. These theorems appear frequently in GCSE maths exams and once you understand the patterns, they become powerful problem-solving tools.

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Basic Circle Theorems

Quick tip: Remember "centre doubles circumference" - the angle at the centre is always double the angle at the circumference for the same arc.

Finally, any angle in a semicircle is always 90°. If you spot a triangle where one side is the diameter of a circle, you've found yourself a right-angled triangle.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Circle Theorems

Exam strategy: These theorems often combine in questions, so look for multiple rules working together rather than just one isolated theorem.

Understanding these patterns means you'll recognise the theorems quickly in exam questions. Practice identifying which theorem applies first, then apply the rule confidently.