Inversion and Spiral

Inversion and Spiral is basically what it sounds like: a combination of the units Invert and Spiral, though admittedly there is a lean towards the latter. It is a pretty fun unit, and not too difficult for a Z if you are a geo main and have grasped the concepts well. Admittedly, though, you do get a large hint for each PSET problem (to invert or to find a spiral similarity!), though it is harder to decide between the two than you may think. The Queue point (Orthic Miquel) and Sharkydevil point (Intouch Miquel) feature quite a bit here, if you are a config lover.

Should I do this unit?

Of course: It's a great unit :). You will have an easier time if you are relatively familiar with EGMO Chapters 8 and 10, or have done either/both of the Invert and Spiral units.

Philosophy

Most problems in this unit revolve around either finding a suitable point to invert at (and inverting there) to simplify the problem, or finding some Miquel point/spiral center and using properties obtained from the spiral similarity (including obtaining collinearities/concyclicities from the Miquel point config). It is recommended that you have some experience with root bc inversion, inversion about the circumcircle, and inversion about the incircle (knowing which point map to where), and be able to identify the Queue and Sharkydevil points if they pop up.

Notable Problems

1. 2016 IMO 3: A very cool walkthrough which is an unexpected use of inversion; understand the geo part thoroughly.
2. 2012 ISL G6: An elegant spiral similarity walkthrough which is extremely nontrivial to come up with; expect to spend a bit of time here too.
3. 2018 Kazakhstan: Spend some time on this one; it will give you great satisfaction if you solve it :).
4. 2014 Taiwan TST, Cosmin Pohoata: Do your best on this one; its hard, but very rewarding and a great example. Required on ZGY.
5. 2016 ELMO 2: This is another fairly hard but elegant problem.
6. 2018 IMO 6: This is... a problem of all time.