I'm afraid I can't provide the pictures to the corresponding properties... but I believe the syllabus itself actually has a very good section detailing all of them.
You should be able to infer the abbreviations from most of them.
Chord Properties
1. The perpendicular from the centre of the circle to the chord bisects the chord, and conversely, the line joining the centre of the circle to the mid-point of a chord is perpendicular to the chord.
2. Chords in a circle that are equidistant from the centre are equal, and conversely, equal chords in a circle (or in equal circles) are equidistant from the centre.
3. Equal angles at the centre of a circle stand on equal chords, and conversely, equal chords subtend equal angles at the centre of a circle.
4. The products of the intercepts of two intersecting chords are equal.
Angle Properties
5. The angle at the centre is twice the angle at the circumference subtended by the same arc.
6. Angles in the same segment are equal
or
Angles at the circumference subtended by the same arc are equal.
7. The angle at the circumference in a semi-circle is a right angle.
Cyclic Quadrilaterals
8. Opposite angles of a cyclic quadrilateral are supplementary (i.e. sum to 180 degrees).
9. If the opposite angles of a quadrilateral are supplementary, the quadrilateral is cyclic.
10. The exterior angle to a vertex of a cyclic quadrilateral equals the interior opposite angle.
11. If an interval subtends equal angles at two points on the same side of it, the end points of the interval and the two points are concyclic.
Tangent Properties
12. The tangent is perpendicular to the radius through the point of contact.
13. Two tangents drawn to a circle from an external point are equal in length.
14. The angle between a tangent and a chord is equal to the angle in the alternate segment ("alternate segment theorem").
Note - this is a very important and powerful theorem that is often required in exams.
15. The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point.
Two-Circle Properties
16. If two circles touch (externally or internally), the line joining their centres passes through their point of contact, i.e. the centres and the point of contact are collinear.
Property of Non-Collinear Points
17. Any of three non-collinear points lie on a unique circle whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining the points.
- HSC Pocket Formula Book, Maths Extension 1
