Circle Theorems and Circle Proofs
Circle theorems turn a diagram into a chain of reasons. Learn each relationship, why it is true, and how to combine it with triangle angles, congruence and Pythagoras.
Higher. Review angles, congruence, circle vocabulary and Pythagoras first.
Start with the circle and its centre
A circle theorem is a justified relationship between angles or lengths. O denotes the centre. Radii are equal, so a triangle formed by two radii is isosceles. A chord joins two circumference points; a tangent touches at one point. Mark what is given before applying a theorem.
Angle at the centre and angle in a semicircle
The angle at the centre is twice the angle at the circumference standing on the same arc. Select the central angle matching the arc: it may be reflex. If ∠APB=38°, the matching ∠AOB=76°.
Proof of the centre theorem
Join OP. In the diagram, let ∠APO=x and ∠OPB=y. Since OA=OP, ∠AOP=180°−2x. Since OB=OP, ∠POB=180°−2y. Angles around O total 360°, so the remaining central angle AOB is 360°−(180°−2x)−(180°−2y)=2(x+y)=2∠APB. If O lies outside the two arms at P, use the difference of the two isosceles-triangle angles instead of their sum; the same double-angle result follows for the matching arc.
An angle in a semicircle is 90°. Its chord is a diameter, so the matching central angle is 180°; half is 90°. Example: if AB is a diameter and triangle APB has angle A=32°, then P=90° and B=58°.
Same segment and cyclic quadrilaterals
Angles in the same segment are equal. They stand on the same chord, with their vertices on the same side of it. Proof: each is half the same central angle. If ∠APB=47°, another circumference angle AQB standing on chord AB in the same segment is 47°.
Opposite angles in a cyclic quadrilateral sum to 180°. All four vertices must lie on the circle. Proof: the two opposite angles stand on the two complementary arcs joining the other two vertices; their matching central angles total 360°, so the circumference angles total 180°. Thus if ∠A=112°, opposite ∠C=68°.
An exterior angle of a cyclic quadrilateral equals its opposite interior angle: each is 180° minus the adjacent interior angle. Do not use the cyclic rule on an arbitrary quadrilateral.
Radii, tangents and equal tangent lengths
A tangent is perpendicular to the radius at the point of contact. To understand why, drop the perpendicular from O to the tangent line. If its foot differed from the contact point T, that perpendicular distance would be less than OT, so the line would pass through the circle instead of just touching it. Therefore the perpendicular meets the line at T.
The two tangents drawn from one external point have equal lengths. If they touch at A and B, join OA, OB and the external point X to O. Right triangles OAX and OBX have equal radii and the common hypotenuse OX, so RHS congruence gives XA=XB.
Worked example
Tangents XA and XB meet at 50°. In quadrilateral OAXB, the contact angles are both 90°. Therefore ∠AOB=360°−90°−90°−50°=130°. If XA=7 cm, XB is also 7 cm.
Chords and perpendicular bisectors
A perpendicular from the centre to a chord bisects that chord. Let M be its foot on chord AB. Right triangles OMA and OMB share OM and have equal hypotenuses OA and OB. RHS gives AM=MB.
Conversely, a line from the centre through the midpoint of a non-diameter chord is perpendicular to it: SSS makes the two triangles congruent, so the equal adjacent angles at M must each be 90°.
Worked example
A chord has length 16 cm in a circle of radius 10 cm. Half-chord=8 cm. The perpendicular distance d from O to the chord satisfies d²+8²=10², giving d=6 cm.
Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment. The angle at the circumference must stand on that chord on the opposite side from the tangent–chord angle being used.
Why it works
For a minor chord subtending central angle 2θ, the isosceles triangle formed by its radii has each base angle 90°−θ. The tangent is perpendicular to the radius, so its angle with the chord is 90°−(90°−θ)=θ. The circumference angle standing on that chord in the alternate segment is also θ by the centre theorem. Choosing the supplementary tangent angle gives the supplementary circumference angle in the other segment.
A complete geometric argument
Suppose AB is a diameter, C lies on the circle, and the tangent at A makes 35° with chord AC in the angle opposite ∠ABC. Alternate segment gives ∠ABC=35°. The semicircle theorem gives ∠ACB=90°. Triangle angle sum then gives ∠BAC=55°. Name the theorem at every step; a bare list of numbers is not a proof.
Think Like a Mathematician
Draw a radius that is not already shown. It often creates an isosceles or right triangle, allowing ordinary angle facts to unlock a circle problem.
Common Mistakes
The centre must be identified; it is not any interior point. Same-segment angles must stand on the same chord on the same side. Opposite angles sum to 180° only for a cyclic quadrilateral. A tangent is perpendicular to the radius at contact, not to every chord.
Practice — circle theorems and reasons
1. A central angle is 124°. Find a circumference angle standing on the same arc.
124÷2=62°, by the centre theorem.
2. AB is a diameter. Triangle ACB has angle A=27°. Find C and B.
C=90° (semicircle); B=180−90−27=63°.
3. One angle of a cyclic quadrilateral is 103°. Find the opposite angle.
180−103=77°.
4. Two angles stand on the same chord in the same segment. One is 58°. Find the other.
58°, angles in the same segment are equal.
5. Tangents from X touch at A and B. XA=11 cm. Find XB and angle OAX.
XB=11 cm (equal tangents), and ∠OAX=90° (radius and tangent).
6. Radius 13 cm, chord length 24 cm. Find the distance from centre to chord.
Half-chord=12. Distance=√(169−144)=5 cm.
7. A tangent–chord angle is 64°. Find the angle in the alternate segment.
64°, by the alternate segment theorem.
8. Give a proof that tangents from one external point are equal.
Join the centre to both contact points and the external point. The two right triangles have equal radii and a common hypotenuse, so are congruent by RHS. Corresponding tangent lengths are equal.
Frequently Asked Questions
Must I prove every theorem whenever I use it?
No. State the theorem as a reason unless the question specifically asks for its proof. You must understand the proofs and be able to construct related arguments.
Can the central angle be reflex?
Yes. Match the arc used by the circumference angle; using the other central angle gives the wrong result.
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