Yeah I hated circle geometry too when I was in Year 11 and Year 12. Here's a few pointers that helped me but I don't know if it'll help much for you since they're pretty basic:
- Always go 'angle hunting' whenever you see a big 3 or 4 mark circle geometry question. Normally, the tougher questions require really obscure information and sometimes you won't be able to recognise them without knowing the values of certain angles. Also, knowing as many angles as possible can help you identify the presence of perhaps alternate angles (when proving that two lines are parallel) or maybe when proving that there is a cyclic quadrilateral. My advice is to go angle hunting for all circle geometry questions to begin with if you have the time (even if the questions are only like 1 or 2 marks).
- This tip is really basic but I legit just learned this like at the end of Year 11 which was pretty sad. So you know how cyclic quadrilaterals are when opposite angles of a quadrilateral (whose corners are touching the circumference of the circle) are supplementary (I think)? I didn't realise that the 'angles in the same segment' theorem can also be used to show a cyclic quadrilateral. This can be pretty handy in exams.
- Whenever you see a 'butterfly' looking shape in a circle like this
http://www.timdevereux.co.uk/maths/geompages/geompics/proof3_a.gif , then the question will almost always require angle in the same segment theorem. This is really useful especially when the diagram becomes a bit more convoluted like this
http://community.boredofstudies.org...88568-circle-geometry-question-circlegeom.png
(note how we can use this theorem for triangle DAB and triangle DPC)
- Whenever you see a tangent, then there's probably gonna be the 'angle in alternate segment' theorem.
Anyways I don't know if these tips helped - you might already know all of this but it helped me personally for my HSC.
In terms of structuring response, just list out all of your steps as usual (like proving similar triangles). Just write your reason in brackets (like angle in same segment or whatever) and always state what side or angle you're referring to.
I would recommend copying to diagram in your writing exam booklet if you have the time. Also, if you need to draw another line in the diagram, just say 'construct line DC' or something like that.
Finally, the best way to master circle geometry is through practice. There's really no shortcut except doing many questions. And also, I would recommend looking at the circle geometry questions during reading time of an exam - you can normally figure out how to prove something in your head for the simpler circle geometry questions.
Good luck!
