Circle theorems

What you are looking at

Each tab is one theorem, stated at the top and drawn below: angle at the centre, angles in the same segment, angle in a semicircle, cyclic quadrilateral, tangent and radius, alternate segment, tangents from a point, and the perpendicular from the centre to a chord. The points are draggable and snap round the circle; the working on the right gives every angle and the relation the theorem claims, so you can check it holds at every position, not just the one in the textbook diagram.

The tool also shows where the statements have to be read carefully. Drag $P$ onto the minor arc and the angle at the centre becomes the reflex angle $AOB$ — the working switches with it. Move $Q$ into the other segment and the two angles on the chord are no longer equal; they are supplementary instead. Play moves one point round the circle for you, which is the quickest way to see that a theorem is about every position at once.

Try this

1. 1On Angle at the centre, drag $P$ all the way round. Angle $APB$ stays half of angle $AOB$ — and when $P$ crosses onto the minor arc, it is half of the reflex angle.
2. 2On Semicircle, drag $B$: the diameter turns with it, and angle $APB$ reads $90^\circ$ wherever $P$ is.
3. 3On Cyclic quadrilateral, drag any vertex and watch both sums: angle $DAB$ + angle $BCD$ and angle $ABC$ + angle $CDA$ each stay $180^\circ$.
4. 4On Tangents from a point, pull $P$ further from the circle: $PA$ and $PB$ stay equal, and $OP$ still bisects the angle between the tangents.

Mistakes students make

• Doubling the wrong angle. When $P$ is on the minor arc, angle $APB$ is half the *reflex* angle at the centre, not half the angle drawn inside the triangle.
• Pairing the alternate-segment angles wrongly. The angle between the tangent and the chord $TA$ equals the angle in the segment on the *other* side of the chord — angle $TBA$ — not the one next to it.
• Using the semicircle theorem when $AB$ is not a diameter. The right angle only appears when the chord passes through $O$; a chord that misses the centre gives the same-segment theorem instead.

In the exam

• A circle-theorem question is marked on reasons as much as angles: write the theorem's name at each step (“angle in a semicircle is $90^\circ$”, “opposite angles of a cyclic quadrilateral sum to $180^\circ$”). A correct number with no reason loses the reason mark.
• Before calculating, mark what the diagram gives you: two radii make an isosceles triangle, a tangent meets its radius at $90^\circ$, and a diameter is the hint for the semicircle theorem.

Where it is taught

• Cambridge IGCSE Mathematics 0580/0980 · 4.5 Circle theorems (Core: angle in a semicircle, tangent and radius · Extended: the rest)