This is a couple of lessons on circle theorems that was designed for able KS4 students.  

In lesson 1, we discussed different ways of defining a circle (how many ways can you think of?) and progressing onto exploring the idea that isosceles triangles are central to deriving many of the circle theorems by looking at geogebra applets like this. 
  
Lesson 2 was inspired by the amazing version of Euclid's elements written by Oliver Byrne in 1847, which has recently been reprinted.  Students studied copies of the some circle theorems (the ones derivable from isosceles triangles - which ones are they?), then were asked to reproduce their own versions of the proofs in pairs - see the slide show below for a sample of the results!  This was extremely successful in demanding that students discuss and understand the logical structure of the arguments.