Mathematical Notes

How to start writing olympiad proofs

A circle problem shows how to turn something you can see into something you can justify.

Start with the claim

Let AB be a diameter of a circle with centre O. Let C be any other point on the circle. Prove that angle ACB is a right angle.

The drawing suggests a right angle. To prove it, look for something you know exactly. Here O is the centre, so OA, OB and OC are all radii and have the same length.

Triangle ABC is inscribed in a circle. AB is a diameter through centre O. — Figure 1. Start with the angle at C.

Make the radii do some work

Draw OC. It splits triangle ABC into two isosceles triangles, AOC and BOC. Call angle CAO α and angle CBO β.

OC splits ABC into two isosceles triangles. Alpha and beta label their equal base angles. — Figure 2. OC forms two isosceles triangles with equal base angles.

Because OA = OC, angle ACO is α. Because OB = OC, angle OCB is β. Thus angle ACB is α + β. Since O lies on AB, the angles at A and B of triangle ABC are α and β.

Now add the three angles of triangle ABC. We get α + β + (α + β) = 180°. Therefore α + β = 90°, which is what we wanted to prove.

Write what you need to prove. List one fact that follows from the givens. After each step, ask which fact or theorem justifies it.