To prove this, you need to split the quadrilateral up into 4 triangles, by drawing lines from the circle centre to the corners. Because each line we draw is a radius of the circle, this means that the four triangles we’ve formed are all isosceles triangles. This means that the outer angles in each triangle are the same, so we can label them to show this:
Now, we know that the sum of the interior angles of a quadrilateral is 360°. So we can write down an equation:
Notice that ‘e + f’ together make up the angle in one corner of the quadrilateral, and that ‘g + h’ make up the angle in the oppositecorner of the quadrilateral. We can rearrange the equation to show this a bit more clearly:
Now what about the other two corners – we need to prove that their angles add up to 180° as well. One corner is made up of ‘h’ and ‘e’, and the other corner is made up of ‘f’ and ‘g’. We can do this by rearranging the equation in a slightly different way:
This shows that the opposite angles of a cyclic quadrilateral are supplementary.
