Overview

Olympiad geometry and AIME geometry are quite different. This section will be a short introduction to olympiad geometry, including constructions in geometry.

Before diving into explanation, let us introduce famous notation and words first:

• Three or more points are collinear if it lies on a single line. If $P_1, P_2, \dots , P_n$ are collinear then the line is often denoted as $\overline{P_1P_2\dots P_n}$.
• Four or more points are concyclic if it lies on a single circle. If $Q_1, Q_2\dots Q_m$ are concyclic then the circle contain those points are often denotes as $(Q_1Q_2\dots Q_m)$.
• The intersection of two objects: intersection of $\ell$ and $m$, possibly line or a circle, is denoted as $X=\ell \cap m$.

Note that those notations are not so common (though used quite often) so I recommend to define it before using during the contests.

About olympiad geometry

Olympiad geometrys are roughly divided to three flavors:

1. Synthetic, which uses only elementary facts about configurations and it doesn't contain heavy computation. This includes,

   • Angle chansing,
   • Power of a point,
   • Homothety,
   • Sometimes inversion is notes as synthetic
2. bashing, which is a heavy calculation to automatically (doesn't mean easy!) finish a problem with a good setup. This includes,

   • Coodinates bashing (though rare on olympiad),
   • Complex bashing,
   • barycentric coordinates.
3. Projective, which is about cross ratios and harmonic bundles (it's okay you don't understand it for now). This includes,

   • Harmonic bundles,
   • Projective transformation,
   • moving points.

It is possible that a single problem have two or more solutions with a different type written above. (In fact every problem is coordinate bashable.) But since you have three problems on 4.5 hours during a contest, it's not a good idea to solve problem with coordinates which have many circles, or complex bash without a poor choice of unit circle, etc. You'll need to find a natural way to prove the statement.

For more description, read Evan Chen's article: https://web.evanchen.cc/faq-contest.html#C-14

About constructing diagrams

This section is based on: https://web.evanchen.cc/handouts/Constructions/Constructions.pdf

Constructing diagram is important for any kind of geometry. The reason it's important is that:

• finding key observations. Points $A, B, C$ are collinear, or $W, X, Y, Z$ are concyclic, etc.
• Thinking how to write a diagram is sometimes part of the problem. We'll cover this in an example problem later.

tips:

• Buy a good ruler and compass, with ruler having perpendicular lines; this will save so much time while having a clear accurate diagram.
• In a triangle geometry (which is the problem that starts from "Let $\triangle{ABC}$ be a triangle) always start writing circle $(ABC)$ first, even if you don't use it at all. It helps you write many point and triangle centers with only using a ruler.
• Some students prefer using colored diagrams, which highlights important circles and lines well.
• Try not using compass as you can.

Writing diagrams using computer

There's two digital system called Geogebra and Asymptote. We won't dive in deeply but if you're curious read evan's article above.

Example problem

TODO

Practice Problems