Core Circle Theorems
Every circle theorem explained with WHY it works, how to spot it in a diagram, and full worked proof examples — the foundation for 40+ marks in Paper 2.
Circle Geometry
Table of Contents
Circle Geometry: Theorems, Proofs & Reasoning#
Circle geometry is the biggest new topic in Grade 11 and one of the highest-weighted sections in Paper 2 (~40 marks). Unlike algebra where you calculate, here you must prove — and every line of your proof needs a reason from the approved list of theorems.
The Golden Rule: Radii Create Isosceles Triangles#
Almost every proof starts here: all radii of a circle are equal. Whenever you see the centre $O$ connected to two points on the circle, you have an isosceles triangle. That gives you two equal base angles — which is usually the key to unlocking the rest of the proof.
The Three Theorem Families#
Family 1: Centre & Chord Theorems#
| Theorem | Statement | Exam reason |
|---|---|---|
| Perpendicular from centre | Line from centre ⊥ to chord bisects the chord | line from centre ⊥ to chord |
| Angle at centre | Angle at centre = $2 \times$ angle at circumference | $\angle$ at centre = $2 \times \angle$ at circum |
| Angle in semicircle | Angle subtended by diameter = $90°$ | $\angle$ in semi-circle |
| Equal chords | Equal chords subtend equal angles | equal chords, equal $\angle$s |
Family 2: Angles on the Circle#
| Theorem | Statement | Exam reason |
|---|---|---|
| Angles in same segment | Angles subtended by the same arc are equal | $\angle$s in same seg |
Family 3: Cyclic Quadrilaterals & Tangents#
| Theorem | Statement | Exam reason |
|---|---|---|
| Opposite angles | Opposite angles of a cyclic quad add to $180°$ | opp $\angle$s of cyclic quad |
| Exterior angle | Ext angle of cyclic quad = interior opposite angle | ext $\angle$ of cyclic quad |
| Tangent ⊥ radius | Tangent is perpendicular to radius at point of contact | tan ⊥ rad |
| Two tangents | Two tangents from same external point are equal | tans from same pt |
| Tan-chord angle | Angle between tangent and chord = angle in alternate segment | tan-chord theorem |
The Proof Strategy#
For every circle geometry problem:
- Mark the diagram: Draw all radii, mark equal lengths, mark right angles.
- Identify isosceles triangles: Radii → equal sides → equal base angles.
- Identify the theorem: Match the configuration to a theorem from the table above.
- Write statement + reason: Every line of your proof needs BOTH.
Deep Dives (click into each)#
- Core Circle Theorems — the 5 fundamental theorems with proofs, worked examples, and the thinking process
- Cyclic Quadrilaterals, Tangents & Proofs — extending the theorems to four-sided figures and tangent lines
🚨 Common Mistakes#
- Wrong reasons: Using “angles in a circle” instead of the SPECIFIC theorem name. Markers follow a strict list — use the exact wording from the table above.
- Assuming the centre: Never assume a point is the centre unless the question says so. Without the centre, you can’t use “angle at centre” theorem.
- Forgetting the reflex angle: When a point is on the minor arc, the centre angle is the REFLEX angle ($360° - \theta$).
- Not marking the diagram: Before writing anything, mark ALL radii, right angles, and equal angles. This reveals which theorems apply.
🔗 Related Grade 11 topics:
- Analytical Geometry: Circles & Tangents — the algebraic approach to circles
- Quadratic Equations — algebraic skills needed for some proofs
📌 Grade 12 extension: Euclidean Geometry — proportionality, similarity, and the Pythagorean proof
⏮️ Trigonometry | 🏠 Back to Grade 11 | ⏭️ Analytical Geometry
Cyclic Quads, Tangents & Proofs
Master cyclic quadrilateral properties (and how to PROVE a quad is cyclic), tangent theorems, the tan-chord angle, and multi-step geometry proofs — with full worked examples and exam strategies.