MODA: Geometry bashing

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.

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.

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