Circle Theorems

I have had a few requests for circle theorems from people I know. While I will be happy to write about this topic, I feel especially at a GCSE level, the circle theorem questions can often be more difficult compared to the rest of the paper. Spotting what theorem to use can be extremely difficult to do, even if you know the theorems inside out. So before I get started, I would like you to know that knowing the theorems is not enough. You need to practice spotting the theorems and getting used to the questions. No teacher on the planet can show you how to do this: they can show you the theorems, but you need to be able to apply them through practice.

There are 9 theorems to know about; I will group them into three categories: the obvious ones, the ones that need learning and the more obscure ones. However, what goes into what category is entirely subjective, so please keep that in mind.

The Obvious Ones:

The radius and the tangent are at 90 degrees to each other, this should seem quite obvious as they are clearly perpendicular to one another.

the green is the radius, and the blue the tangent

Two radii form an isosceles triangle; this should also be obvious as an isosceles triangle has two sides of the same length and the radius is the same all around the circle.

Tangents drawn from the same point meet at the circumference with the same length, this can be hard to understand in words, but I'm basically saying the two tangents below are equal in length from the point they meet to where they each meet the circumference.

the green and black lines are tangents

The Ones That Need Learning:

The angle in a semicircle is 90 degrees. People often forget this one, remember it! However, this is providing the line goes through the centre of the circle (the blue line).

The angle at the centre is twice the size of the angle at the circumference.

Also, angles opposite the same segment are always equal.

The Obscure Ones:

The alternate segment theorem states that the angle between the tangent and the chord is equal to the angle in the opposite segment. It's difficult to explain, but hopefully the diagram helps!

the chord is the black line, the tangent is purple

Cyclic quadrilaterals: both pairs of opposite angles add to 180 degrees.

So:

Now there is another theorem, stating that the bisector of a chord passes through the centre of the circle: this is simple, however, providing you understand that the bisector is the perpendicular line through the mid-point of the chord.

the green line is the chord, and the blue line the bisector-which passes through the centre

I have briefly gone through all 9 theorems, however I haven't discussed any example problems, although I would be happy to in the future. To get the images I used the graphing software desmos. If you would like me to briefly go through anything else I would be happy to, please leave a comment. Thanks for reading-naBla5040