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Geometry

Geometry is one of the most classical areas of olympiad mathematics.
Much of the basic toolkit coincides with the synthetic geometry studied in the first two years of high school, but olympiad problems often involve additional techniques and theorems.

In olympiad problems it is essential to learn to observe figures carefully, recognize known configurations, and connect different geometric properties.
Techniques such as angle chasing (tracking relationships between angles) and systematic use of similarity often lead to the solution.

Below are some of the main topics to know.

Fundamental topics

  • Basic geometry
  • Configurations and theorems
  • Transformations and techniques

Triangles and quadrilaterals

A very large portion of olympiad problems relies on properties of triangles and quadrilaterals.

Among the fundamental results are:

  • triangle congruence criteria
  • similarity criteria
  • properties of special quadrilaterals

These tools allow comparisons between lengths, angles, and areas.

Pythagorean theorem and Euclid’s theorems

The Pythagorean theorem and Euclid’s theorems on right triangles are among the most frequently used results.

They often appear in combination with similarity and proportional relationships between segments.

Circles

Circles are one of the most common configurations in olympiad problems.

It is important to know:

  • central and inscribed angles
  • properties of chords and tangents
  • conditions for a quadrilateral to be cyclic

Many problems reduce to recognizing that certain points lie on the same circle.

Power of a point

The power of a point with respect to a circle relates the lengths of chords, secants, and tangents.

It is a powerful tool for proving equalities between segments.

Cevians and triangle centers

In triangles, special segments called cevians (medians, altitudes, angle bisectors) often appear.

From these arise many notable points of the triangle, such as the centroid, orthocenter, and incenter.

Knowing their properties helps quickly recognize recurring geometric configurations.

Classical theorems

Several classical theorems frequently appear in olympiad problems, including:

  • Ceva’s theorem
  • Menelaus’ theorem
  • Ptolemy’s theorem
  • Pascal’s and Pappus’ theorems

These results connect ratios of segments and properties of more complex geometric configurations.

Geometric transformations

Many problems become simpler by viewing the figure through plane transformations.

The most important are:

  • isometries (translations, rotations, reflections)
  • homotheties
  • similarities
  • affine transformations

Each transformation preserves certain properties (invariants) that can be exploited in proofs.

Angle chasing

Angle chasing is a widely used technique in olympiad geometry.

It consists of systematically following angle relationships within a figure to deduce new equalities and identify known configurations.

Vectors in the plane

In some problems it is useful to use vectors in the plane, which allow geometric configurations to be handled using algebraic tools.

Where to start preparing

The main prerequisite for tackling olympiad geometry is a solid knowledge of synthetic geometry from the first two years of high school.

When studying geometry, do not limit yourself to memorizing theorem statements: it is essential to understand and study the proofs.
Only by seeing many different proofs can one learn to develop new ones.

School textbook

The first resource to use is the geometry textbook from the first two years of high school, which covers most of the theoretical foundations required.

Senior Pills

To go deeper and reach a more advanced level, you can consult the Senior Pills, a collection of video lectures from the Senior training camp of the Mathematics Olympiads.

🎬 http://olimpiadi.dm.unibo.it/il-senior-in-pillole/

Further information about the Senior training camp is available on the page Senior Training Camp Lessons.

For further study, several volumes from the U Math series are very useful:

In the Resources section of the website you can find further information about the U Math series and other useful materials for training.