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Cyclic Quadrilaterals

Age 11 to 16
Challenge Level

If you prefer to work on paper, you may wish to print out some sheets of circles:
9 dot 10 dot 12 dot 15 dot 18 dot


In the GeoGebra interactivity below there is a circle with 9 equally spaced points on the edge, and one in the centre.

Draw as many different triangles as you can, by joining the centre dot and any two of the dots on the edge.



Can you work out the angles in your triangles?

 

 


You should have found four different triangles with angles of:
40, 70, 70
80, 50, 50
120, 30, 30,
160, 10, 10
 

 


Now draw a few quadrilaterals whose interior contains the centre of the circle, by joining four dots on the edge.

Can you work out the angles of your quadrilaterals?

If you're finding it hard to work out the angles, take a look at Getting Started.

Create at least five different quadrilaterals in this way and work out their angles.



What do you notice about the angles on opposite vertices of your quadrilaterals?

Perhaps you are wondering whether this only happens with 9-dot circles...

You may wish to explore the opposite angles of quadrilaterals on circles with a different number of dots.

Click below for interactivities with 10, 12, 15 and 18 dots around the circle.

10 dots

 

 


12 dots

 


15 dots

 


18 dots

 


Extension:

Will the same happen if you draw a circle and choose four points at random to form a quadrilateral?


To prove that the opposite angles of all cyclic quadrilaterals add to $180^\circ$ go to Cyclic Quadrilaterals Proof


Quadrilaterals whose vertices lie on the edge of a circle are called Cyclic Quadrilaterals.


You may be interested in the other problems in our Sharpen your Skills Feature.