Circle Properties

1. Angle at centre is twice the angle at circumference (same arc).

If a chord subtends an angle of $\alpha$ at the centre, it subtends $\alpha/2$ at any point on the major arc.

2. Angle in a semicircle is $90°$.

(Special case of theorem 1: if the central angle is $180°$, the inscribed angle is $90°$.)

3. Angles in the same segment are equal.

Two chords from the same arc subtend the same angle wherever measured on the major arc.

4. Opposite angles in a cyclic quadrilateral sum to $180°$.

A 4-vertex inscribed shape has opposite angles supplementary.

5. Tangent is perpendicular to radius at the point of contact.

If a line touches the circle at point $P$, and $O$ is the centre, then $OP \perp$ tangent.

6. Two tangents from an external point are equal in length.

If two tangents are drawn from outside the circle, the segments from external point to tangent points are equal.

7. Alternate segment theorem.

The angle between a tangent and a chord = the angle in the alternate segment (the segment on the other side of the chord).

Worked qualitative. Why is the angle in a semicircle $90°$?

• Diameter creates a central angle of $180°$.
• Inscribed angle: $180/2 = 90°$.
• This is theorem 2 — special case of 1.

Edexcel tip. ALWAYS state the theorem name. 'Angles in the same segment' or 'angle at centre is twice angle at circumference' — exact phrasing.

Method:

1. Identify what features are present (tangent? chord? cyclic quadrilateral?).
2. Match to a theorem.
3. Apply.

Common configurations:

Feature	Theorem
Tangent at $P$	Tangent ⊥ radius (#5)
Two tangents from $P$	Equal lengths (#6)
Inscribed quadrilateral	Opposite angles supplementary (#4)
Diameter as chord	$90°$ angle elsewhere (#2)
Chord with two angles on either side	Equal in same segment (#3)
Tangent and chord meeting	Alternate segment (#7)

Worked example. Tangent at $P$, chord $PQ$, angle between them $40°$. Find angle subtended by $PQ$ at the far side.

• Alternate segment theorem.
• Far side angle = $40°$.

Multi-step problems. Often you'll combine 2-3 theorems in one problem.

Worked qualitative. Why are circle theorems useful?

• Without them, you'd need to set up triangles and use long Pythagoras / trig calculations.
• With them, many angles can be found directly.
• They embody beautiful geometric symmetries.

Edexcel tip. Mark schemes for circle problems award:

• M1 for identifying the configuration.
• M1 for naming the theorem.
• A1 for the angle.

Always state the theorem!