Cyclic quadrilaterals - Higher

A cyclic quadrilateral is a quadrilateral drawn inside a circle. Every corner of the quadrilateral must touch the circumference of the circle.

Cyclic and non-cyclic quadrilateral

The second shape is not a cyclic quadrilateral. One corner does not touch the circumference.

The opposite angles in a cyclic quadrilateral add up to 180°.

a + c = 180^\circ

b + d = 180^\circ

Cyclic quadrilateral with angles a, b, c and d

Example

Calculate the angles a and b.

Cyclic quadrilateral with angles a, b, 60 degrees and 140degrees

The opposite angles in a cyclic quadrilateral add up to 180°.

b = 180 - 140 = 40^\circ

a = 180 - 60 = 120^\circ

Proof

Let angle CDE = x and angle EFC = y.

Cyclic quadrilateral (angles x and y at the circumference)

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

Angle COE = 2y and the reflex angle COE = 2x.

Cyclic quadrilateral with angles x and y at the circumference and 2x and 2y at the centre

Angles around a point add up to 360°.

2y + 2x = 360^\circ

\frac{2y}{2} + \frac{2x}{2} = \frac{360^\circ}{2}

So y + x = 180^\circ