Invert

Although it looks scary because it is placed so far down on the unit list, Invert is one of the most straightforward units to grasp theory-wise. With its short and sweet two-syllable title, Invert is the epitome of simplicity and elegance in olympiad geometry. Whether you are a geo fanatic or not, Invert will instantly capture your heart, and make you wish every geo problem was trivial by inversion!

Philosophy

One nice thing is that every problem on the Invert unit will use inversion, and that will be the main part, if not all, of it. Most of the problems are just a combination of inversion and angle chasing. So it's like you start every problem off with one hint already. Then the hard part is just figuring out which point to invert from. Usually, there will be only a few possible options, since this point ideally is the intersection of many lines/circles, and is not too obscure of a point.

Additionally, many of the problems involve inverting from a point, which may seem unnatural to one who thinks of "inversion" as inverting a bunch of points and lines about a circle. "Inverting about a point" is simply inverting about an arbitrary circle centered at that point. It doesn't matter which circle, because angles and ratios of segments will be the same no matter which circle one chooses.

One last thing to stress is the importance and underratedness of force-overlaid inversion, which is only briefly mentioned in EGMO but will do most of the work for you in many of these problems. If you want to show that two lines are reflections over the angle bisector of $\triangle{ABC}$, then force-overlaid inversion is probably the way to go.

Also, inversion preserves angles, so don't be afraid to spam a lot of angle chasing!

Some quick things to note:

• Inverting about the incircle, the circumcircle inverts to the nine-point circle of the contact triangle.
• Under force-overlaid inversion, the excircle inverts to the mixtilinear incircle.

Notable Problems

All of these problems have extremely short and slick solutions, and by matching points & clines with their inverses, the problem slowly comes together.

• IMO 2015 P3 A classic example of focusing only on the relevant portion of the diagram, and then inverting.
• No Source Homothety or inversion, whichever shall it be?
• RMM 2018 P6 A very unique problem, but extremely satisfying and approachable.