Spiral

Spiral is a D level unit about, unsurprisingly, spiral similarities and Miquel points. These two concepts show up very frequently in all types of geometry problems, ranging from easy to hard, Config Geo to Weird Geo. Together with Invert, these concepts can be thought as the bread and butter of olympiad geometry. If you've done both, or are feeling adventurous, try doing Inversion and Spiral, a Z unit which combines both topics into one unit. In terms of difficulty, this unit is fairly middle of the road for D units, but there are a couple tricky problems.

Note: if you've done all the problems in EGMO chapter 10, then there should be no new content for you, so you don't really need to request this unit. But it is a good practice set of problems to brush up on geo if you are rusty.

Content

Say you have a quadrilateral $ABCD$. If you intersect opposite sides, you get two more points, $P=\overline{AB}\cap\overline{CD}$ and $Q=\overline{BC}\cap\overline{DA}$. You can think of these as the intersection of the diagonals of the self-intersecting quadrilaterals $ACBD$ and $ADBC$. It turns out that 4 circumcircles in this diagram concur at a point $M$, called the Miquel point. This point has a plethora of many different properties, the most basic of which is that it is the center of the spiral similarity that sends $\overline{AD}\mapsto\overline{BC}$. If $ABCD$ is cyclic, $M$ has even more nice properties.

[image:5 align:left size:medium] Diagram from EGMO

This configuration is ubiquitous in olympiad geometry for how generic it is. It is like a chameleon, it can change into many different shapes and forms through self-intersecting and even degenerate cases, which can sometimes make it tough spot in the wild. Make sure you know it well!

Advice

Because spiral sim and Miquel points are basically just abstracted away angle chasing, they aren't really that high powered. So you can often solve many problems using a different method like inversion, bypassing the spiral sim.

Notable Problems

• IMO 1979/3: This problem is from 1979, yet it still finds a way to be quite tricky. It is required, so there's no way around the weird geo.
• T6POP: A very short and sweet problem that makes you feel nice after solving it.
• TSTST 2013/4: This is a locus problem, something you don't see very often. It's quite the oddity.
• December TST 2013/3: This is the final example problem in EGMO, and also appears in Config Geo. Nonetheless, you can solve it here as well. But be warned, it's quite nontrivial.