A tried and tested technique for attacking geometry problems is ‘bashing’ them with an algebraic approach. We’ve already seen how projective geometry can be based around a coordinate system (giving rise to areal and projective Cartesian coordinates as special cases). Two-dimensional Cartesian coordinates are more naturally manipulated as complex numbers, which is usually the best general approach if you’re stuck on a problem and don’t know how to progress.

- Preface
- Combinatorics I (enumerative and geometrical combinatorics)
- Linear algebra
- Combinatorics II (graph theory)
- Polynomials
- Sequences
- Inequalities
- Projective geometry
- Complex numbers
- Triangle geometry
- Areal coordinates
- The Riemann sphere
- Diophantine equations
- Conic sections
- Glossary
- Further reading
- Acknowledgements

The three- and four-dimensional counterparts of the complex number bash are Hamiltonian quaternion bashes, which are more difficult due to the lack of commutative multiplication. I used quaternions, for instance, to construct the discrete Hopf fibration. This is not covered in MODA, as problems in olympiad geometry are almost invariably two-dimensional.