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If the points are A, B, C; find the equations of the perpendicular bisectors of AB and AC (not that hard with either the distance formula or just find midpoint AB and -1/gradient AB, same for AC) and then find the point where they intersect. Call this point O, its the centre of your circle...

The vast majority of HSC geometry problems are either special cases or direct proofs of theorems not in the syllabus; compare last year's q7 from the 4u paper with http://en.wikipedia.org/wiki/Ptolemy's_theorem. Clearly the question is no more or less stupid as a result.

In what sense?

The first question is a direct consequence of http://en.wikipedia.org/wiki/Desargues'_theorem The second question is a well known result, http://en.wikipedia.org/wiki/Radical_axis. To do part (iv): Let AB and CD intersect at X; it remains to prove X lies on EF. From part (iii), the following...