21.4 Proof of circle theorems — Learn

Prerequisites: use circle angle theorems, triangle angle sums and isosceles triangle facts.

Objectives: apply and prove circle theorems concerning angles, radii, tangents and chords, then use them to prove related results.

Build a proof from known facts

A proof is a chain of statements where each step follows from a definition or a known result. Name the reason for every angle or length statement.

Example 1. $AB$ is a diameter. Point $C$ lies on the circumference. Show that angle $ACB$ is $90^\circ$.

The angle at the centre standing on diameter $AB$ is a straight angle: $180^\circ$.

The angle at the circumference is half the angle at the centre.

Therefore $\angle ACB=180^\circ\div2=90^\circ$.

This proves the angle in a semicircle theorem.

Example 2. A perpendicular from centre $O$ meets chord $AB$ at $M$. Show that $AM=MB$.

Radii are equal: $OA=OB$.

The two triangles share $OM$, so $OM=OM$.

Both triangles have a right angle at $M$.

The triangles are congruent by RHS: right angle, hypotenuse and one side.

Corresponding sides are equal, so $AM=MB$.

Watch out. A diagram can suggest a result but cannot prove it. Give a reason for each step, then check that your reasons use stated facts or known theorems.

Progress check

For the chapter problem, the perpendicular from the centre bisects the chord. Combine this with the angle theorem from 21.2 to explain why both points on the same segment see the chord at equal angles.

You should now be able to

• write a connected proof with a reason for each step;
• prove the angle in a semicircle result from the centre-angle theorem;
• prove that a perpendicular from the centre bisects a chord;
• use established circle results to prove a related angle or length.