Circle Theorems

Circle Theorems Angles in Same Segment Equal 2

Circle theorems are all about recognizing patterns quickly. Let’s look at an example.

Below is  a circle with a red line cutting through it joining two points on the circle: A and B. This line is called a chord and the little bit of the circle that it chops off at the top is the minor segment.  The big chunk that remains is the major segment. So here’s the first theorem:

  • If we join any point on the circle to A and B this forms a triangle. The angle subtended by the lines from A and B to any point will always be equal.

To illustrate this I’ve drawn two triangles below. See how they make the shape of a butterfly? That’s the first theorem: Angles in the same segment are equal

Angles in the same segment are equal

Circle Theorems Angles in Same Segment Equal

If we move the two points so the butterfly gets a bit squidged notice that the angles are exactly the same, 45 degrees. Bear in mind that in some questions where this theorem applies the chord AB won’t be drawn, but you should still be able to see the butterfly shape.

Circle Theorems Angles in Same Segment Equal 2

Try it out

Slide the two points at the bottom of the circle around and see whether the angles change. Then try sliding those points above the chord into the minor segment at the top. Does the angle change now? Why is that? Are we in another segment? Now try moving the points on the red arc. What’s happened to the angles now?

Angle at Centre Twice Angle at Circumference & Angles in Semicircle

To spot this pattern the key ingredients are:

  • A chord
  • The centre of the circle
  • A point on the circumference

I’m a Star Trek fan and to me this pattern looks like a Starfleet symbol:

Starfleet Command Emblem

Try it out

Slide the two points at the bottom of the circle around and see whether the angles change. What happens if you move the point at the top? To see another theorem slide one of the points so that the blue lines make a diameter and the angle alpha is 180 degrees. What happens to the angle beta between the red lines?

Angles in a Cyclic Quadrilateral

If you see a quadrilateral where all the corners lie on the circle’s circumference, this is called a cyclic quadrilateral, the opposite angles of the quadrilateral add up to 180 degrees.

Try it out

Slide the two points at the top of the circle around and see whether the angles change. See how the red and blue angles always match?

Alternate Segment Theorem

This is probably the most tricky of the circle theorems. It is easy to spot if you recognise three ingredients:

  • A circle
  • A tangent to the circle
  • A triangle inside the circle with one corner lying on the tangent

This  combination means you can relate the angles of the triangle and the angles to the tangent on the opposite side. To see what this means try dragging the points around in the picture below.

Try it out

Slide the two points at the top of the circle around and see whether the angles change. See how the red and blue angles always match?

About the author

ramin

Ramin is a science and maths tutor. He has a first class degree and a doctorate in Physics. He has written two finance books and also teaches adults about investment through his company PensionCraft.