What actually happened at that one competition
In 2019, a geometry problem on the IMO Shortlist appeared deceptively simple. It involved a cyclic quadrilateral with a configuration that most competitors immediately attacked using angle chasing or coordinates. The official solution, when released, used an inversion centered at one of the vertices and a surprisingly clean spiral similarity argument that most people had never considered during the contest. That moment became the reference point for what I'm going to call the Geometry Contest That Rewrote Problem Solving, because it shifted how a lot of people approach competition geometry from that point forward. The core insight is not actually about any single technique. It is about what happens when you draw one diagram and then stop trying to force it into a standard framework. Most solvers, myself included, spend the first twenty minutes of a problem reaching for power circles, spiral similarities, or barycentric coordinates before they have actually looked at what the diagram is telling them. The post-2019 shift was realizing that the diagram itself encodes the solution path if you let it. Here is what I actually do now when I see a geometry problem that looks like it should be hard. I draw the figure to scale using GeoGebra first, not to get an answer but to see where points collapse, where lines become concurrent, and where unexpected symmetries appear. This takes about three minutes. Then I close the software and try to reconstruct the argument from memory. If I cannot recall a specific step, I open it again only for that step. This reduces the cognitive load significantly because the diagram becomes a trusted reference rather than a crutch.
I ran into a concrete edge case last year with a problem involving a complete quadrilateral and a Miquel point configuration. The setup looked standard. I spent twelve minutes chasing angle equalities that led nowhere. The breakthrough came when I dragged one of the free vertices in my GeoGebra file and watched the Miquel point trace a perfect circle through three fixed points. That observation alone collapsed the entire proof into two lines. The diagram had been saying what the algebra could not. The method breaks down in a few specific situations. When a problem involves numerical calculations that depend on exact side lengths rather than projective properties, dynamic geometry software gives you approximations that are sometimes misleading. I once trusted a measured angle in GeoGebra to be exactly sixty degrees when it was actually sixty point two degrees due to a hidden constraint. That wasted about twenty minutes before I caught it. Always verify numerical observations with a proof, not the other way around. Another common failure mode is problems with degenerate configurations where points coincide at critical moments. Software can smooth over these degeneracies by default settings, making the figure look perfectly general when the intended solution actually depends on a limiting case. My workaround is to set the tolerance parameters aggressively low and manually check whether specific points merge at any stage of the construction.
For people who want to practice this approach, the most useful resource is not a textbook but a curated set of past IMO Shortlist geometry problems from 2015 onward. Work through each one using the following sequence. First, draw the figure without any auxiliary lines. Second, identify every cyclic quadrilateral, every concurrent triple of lines, and every pair of similar triangles that appears visually. Third, attempt a proof using only those identified structures. Fourth, if you are stuck, add exactly one auxiliary construction and test whether it creates a new recognizable pattern. This process usually cuts the time spent on a medium-difficulty problem from about forty-five minutes down to roughly fifteen, assuming you already know the basic lemmas. The biggest bottleneck is not the technique itself but the habit of jumping straight to coordinates or trigonometric calculations before exhausting what the pure diagram can tell you. Breaking that habit requires deliberate practice over at least six months of working problems in this sequence. If you are downloading materials or following along with problem sets, make sure you are using the official IMO Shortlist archives or the Art of Problem Solving community problem databases. Third-party compilations sometimes misattribute problem numbers or include problems with incorrect diagrams, which undermines the entire exercise. The files are freely available from the IMO official website under the Shortlist section for each year.
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I still find myself reverting to angle chasing on problems that explicitly resist that approach. The shift is gradual and incomplete. What changed permanently is the default first move, and that alone has been enough to raise my personal average problem completion rate by roughly thirty percent over the last two years of competition preparation.
