Circle Theorems, Constructions & Loci

Parts of a Circle

Before tackling the theorems, you must be fluent with the vocabulary. Every circle question expects you to read a diagram and name the parts instantly.

The radius is a straight line from the centre to the circumference.
The diameter is a chord that passes through the centre; it is twice the radius, so $d = 2r$.
A chord is any straight line joining two points on the circumference.
A tangent is a straight line that touches the circle at exactly one point.
An arc is part of the circumference. The shorter arc is the minor arc, the longer is the major arc.
A sector is the "pizza slice" region bounded by two radii and an arc.
A segment is the region between a chord and the arc it cuts off.

Key terms A cyclic quadrilateral is a four-sided shape whose four vertices all lie on the circumference of one circle.

The phrase "angle subtended" means the angle created at a point by two lines drawn from the ends of an arc or chord. Most theorems are about comparing angles subtended by the same arc.

Circle Theorem 1: Angle at the Centre

The angle subtended by an arc at the centre is twice the angle it subtends at any point on the remaining circumference.

So if the angle at the circumference is $x$, the angle at the centre standing on the same arc is $2x$.

Worked example Points $A$ and $B$ lie on a circle, centre $O$. Point $P$ is on the major arc. The angle $\angle AOB = 130°$. Find $\angle APB$.

Circle Theorem 2: Angle in a Semicircle

The angle in a semicircle is $90°$. This is really a special case of Theorem 1: when $AB$ is a diameter, the angle at the centre is the straight angle $180°$, so the angle at the circumference is half of that.

Spotting a diameter in a problem is your cue to write a $90°$ angle and often reach for Pythagoras or trigonometry inside the right-angled triangle.

Circle Theorem 3: Angles in the Same Segment

Angles subtended by the same arc in the same segment are equal. If two angles both "stand on" chord $AB$ and both sit on the same side, they are identical regardless of where their vertices are.

Exam tip To use this theorem, trace the two lines from each vertex back to the same two endpoints. If both pairs of lines end at $A$ and $B$, the angles are equal. Marking these endpoints clearly is the single biggest time-saver in circle questions.

Circle Theorem 4: Cyclic Quadrilateral

The opposite angles of a cyclic quadrilateral add up to $180°$. They are supplementary.

So $a + c = 180°$ and $b + d = 180°$.

Worked example In cyclic quadrilateral $ABCD$, $\angle A = 95°$ and $\angle B = 70°$. Find $\angle C$ and $\angle D$.

Tangent Theorems

Two key facts deal with tangents:

1. A tangent is perpendicular to the radius drawn to the point of contact. The radius meets the tangent at exactly $90°$.
2. Two tangents drawn from the same external point are equal in length.

The second fact creates an isosceles triangle between the external point and the two contact points, and the line from the external point to the centre bisects the angle between the tangents. This symmetry unlocks many problems.

Watch out Marks are lost when students assume a line is a tangent without justification. A line is only perpendicular to the radius if it is genuinely a tangent (touches once). Check the diagram or wording before applying the $90°$ rule.

The Alternate Segment Theorem

The angle between a tangent and a chord equals the angle in the alternate segment (the angle subtended by that chord from the far side of the circle).

If a tangent touches at point $T$ and a chord $TA$ is drawn, the angle between the tangent and the chord equals the angle that $TA$ subtends at any point on the other side of the chord.

Exam tip This is the hardest theorem to recognise. Look for a tangent and a chord meeting at a single point on the circle. The "alternate segment" is always the one on the opposite side of the chord from the angle you are measuring.

Worked Example: Combining Theorems

Worked example $A$, $B$ and $C$ lie on a circle centre $O$. $TA$ and $TB$ are tangents from external point $T$. The angle $\angle ATB = 50°$. Find $\angle AOB$ and then $\angle ACB$, where $C$ is on the major arc.

Constructions with Compasses

Constructions must be done with a pair of compasses and a straight edge only. Crucially, you must leave all your construction arcs visible — examiners award marks for the arcs, not just the final line.

Perpendicular bisector of a line segment $AB$:

1. Open the compasses to more than half the length of $AB$.
2. With the point on $A$, draw arcs above and below the line.
3. Keeping the same radius, repeat with the point on $B$.
4. Draw a straight line through the two crossing points.

This line cuts $AB$ in half at right angles. Every point on it is equidistant from $A$ and $B$.

Bisector of an angle:

1. Put the compass point on the vertex and draw an arc crossing both arms.
2. From each of those two crossings, draw arcs that meet between the arms (same radius).
3. Draw a line from the vertex through that meeting point.

Every point on this line is equidistant from the two arms of the angle.

Loci

A locus (plural loci) is the set of all points satisfying a rule. The four standard loci are:

Worked example A goat is tied to a corner of a rectangular barn by a rope of length $5\,\text{m}$. Describe the region it can reach.

The locus is an arc of a circle, radius $5\,\text{m}$, centred on the corner, sweeping through the open ground. Where the rope wraps past an adjacent corner, the reachable region becomes a smaller arc whose radius is reduced by the length of that wall.

Real world Loci are how engineers define safety zones and how mobile networks model coverage. The "equidistant from two transmitters" boundary that decides which mast serves your phone is exactly a perpendicular bisector.

When a problem asks for a region rather than a line, find the boundary loci first, then shade the area satisfying every condition at once. Always state whether the boundary itself is included.