You may also like

problem icon

At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

problem icon

No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

problem icon

A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

Cyclic Quadrilaterals

Stage: 3 and 4 Challenge Level: Challenge Level:1

Swaathi from Brighton College Abu Dhabi sent in this excellent solution:

I started this problem by first identifying the different triangles within the 9-dot circle.

Let's figure out the red triangle's angles:

Since a circle has an interior angle of 360 degrees, we can divide 360 by 9 to give us one angle of the triangle. This is because the 9 dots are evenly spaced which means that all the red triangles are identical.


Now, let's look at the green triangle:


If we apply the same technique to the other triangles as well:

Now for the second part of the problem regarding quadrilaterals created using the points on the circumference of the circle.



I think that these methods can be applied to all the circles so in order to prove this theory I tested it with the 10 dot circle.

Extension:


Swaathi claims that opposite angles in any quadrilateral within a circle add up to 180°. Is this enough evidence to prove it?