Euclidean Geometry
Circle theorems and proofs
Circle Theorems and Proofs
Key circle theorems:
• Angle at centre = 2 × angle at circumference (same arc)
• Angle in a semicircle = 90°
• Angles in the same segment are equal
• Opposite angles of a cyclic quadrilateral are supplementary
• Tangent ⊥ radius at point of contact
Example
Applying Circle Theorems
O is the centre. Angle AOB = 100° (at centre).
Angle ACB (at circumference, same arc) = 100° ÷ 2 = 50°.
Cyclic quadrilateral PQRS: angle P = 70°.
Opposite angle R = 180° − 70° = 110°.
Tangent Theorems
• A tangent meets the radius at 90° at the point of tangency.
• Two tangents from an external point are equal in length.
• The angle between a tangent and a chord equals the angle in the alternate segment (tan-chord theorem).
Note
Remember
In proofs, state the theorem you are using as a reason. Draw the diagram clearly and mark equal angles/sides. Practice identifying which theorem applies from the given information.
Key Vocabulary
TheoremA mathematical statement that has been formally proved
Cyclic quadrilateralA quadrilateral whose vertices all lie on a circle
TangentA line that touches a circle at exactly one point
ChordA line segment joining two points on a circle
ArcA portion of the circumference of a circle
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