Circle theorems

April 26, 2022

In this post

Many circle theorems were developed by ancient Greek mathematicians who recognised the use of the circle and looked into its properties in detail.

Construction

To construct a circle we need only use a pair of compasses and set the radius of the circle. This will then generate a perfect circle that avoids the use of drawing by hand, which is rather difficult. We can then use this perfect circle of a set radius to help us work out many different geometrical properties of the circle and other shapes as well.

Combining circles and triangles

When combining circles and triangles we can either have a triangle in a circle such that the points of the triangle are touching the edges of the circle, or we can have a circle inside a triangle so that the sides of the triangle act as tangents to the circle. A tangent is just a line on a circle which is at $90^\circ$ .

To draw a circle around a triangle we must construct two of the perpendicular bisectors of the triangle’s sides to find the centre. Then we simply draw a circle with the same centre and radius from this point to the vertices of the triangle.

To draw a circle within a given triangle we must again use two perpendicular bisectors to find the centre of the triangle. Then we need to construct two of the bisectors of the triangle’s sides and set the radius for our compass as the distance from this bisector to the centre of the circle.

If you need to recap any of the methods for finding bisectors and other methods, please recap the appropriate parts earlier in the course.

Theorems of the circle

A theorem is a statement about something, and when applied to circles we find basic properties which can be very powerful for mathematicians. For your exam you will not be asked to prove any theorems but you may be asked to state some and know what they may be used for. Below shows the main theorems of circles which are used and what they tell us:

Theorem 1

The angle at the centre of a circle is twice the angle at the edge when encompassing a triangle.

Circle theorem angles
This theorem works for any points on the circle for A, B, and P. The angle in the middle (which is actually made by drawing lines to the exact centre of the circle) will always be twice the size of the angle made at the edge (x).

Theorem 2

If a triangle is drawn in a circle with the base going from one side of the circle, through the centre and then to the other, then the angle made will be $90^\circ$ .

As long as one side of the triangle goes through the centre of the circle (shown as the black dot) and the other two sides meet at any other point on the circle (P) then the angle made will be $90^\circ$ .

Theorem 3

Angles in the same segment are equal.

The two angles made from points A and B will always be equal (x).

Theorem 4

Opposite angles in any quadrilateral that is within a circle (with points touching the edge of the circle) equal $180^\circ$ .

For any quadrilateral that is drawn in the above way the opposite angles will total to $180^\circ$ .

So $a+b=180^\circ$ and $c+d=180^\circ$ also.

Using the previous four theorems

These four theorems that we have just learned will probably appear in your exam and so you should know them well and be able to recognise them. It may sometimes be difficult to distinguish between the four, and so you should be very careful when stating properties of a diagram to make sure you are correct. If you are ever unsure, it is good practice to use a protractor to measure certain angles just to make sure; however, in an exam the diagrams may not be drawn to scale so this is not a fool-proof method.