Shabbir Tejani, Year 8, Jack Hunt School, Peterborough, James Dotti and Christiane Eaves, The Mount School York, proved that the opposite angles of a cyclic quadrilateral are equal and also that the exterior angle of the cyclic quadrilateral is equal to the interior opposite angle (Ð TSR = Ð PQR). The proof uses the result, from the January 2000 Six , that the angle at the centre of a circle is twice the angle at the circumference subtended by the same arc.

Here is Shabbir's proof:

Given, a quadrilateral PQRS where vertices lie on a circle, PS is produced to T. Let angle POR = e

To prove that angle c + angle a = 180 degrees (opposite angles of cyclic quadrilateral).

Angle a = ½ angle b (since angle b is at the centre).

Angle c = ½ angle e .

Angle b + angle e = 360 degrees.

Angle a + angle c = 180 degrees.

To prove that angle d = angle a (exterior angle of a cyclic quadrilateral)

Angle c + angle d = 180 degrees (on a straight line).

Angle c + angle a = 180 degrees (opposite angles of cyclic quadrilateral).

Angle d = angle a.